Đề thi toán học không biên giới năm 2016

If B is not an excellent student, then C and D would be excellent students (first statement). However in this case the third statement could not be true. There are two of them.. What i[r]

MATHEMATICS WITHOUT BORDERS 2015-2016

WINTER 2016: GROUP

Problem Which of the following symbols “<”, “>” or “=” should we place in the square, so that the equation would be true?

A) < B) > C) =

Problem What is the missing number?

A) one B) two C) three

Problem If

? =

A) +7 B) -7 C) +2

Problem If

? =

A) B) C)

Problem Replace the smileys with two of the cards in order to get the greatest possible sum

What is the greatest sum?

Problem The number of leaves on a few three-leaf clovers can NOT be:

A) 11 B) 12 C) 15

Problem How many of the following expressions are correct?

A) B) C)

Problem Sonya has fish Amina has fish more than Sonya How many fish Amina and Sonya have in total?

A) B) C)

Problem There is a basket in a dark room In the basket there are yellow and red apples What is the smallest possible number of apples we would need to take out, without looking at their colour, in order to ensure that we have taken out at least red apple?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B 1 + + > –

2 В + = Two is the missing number С 2 + = 6; – = 0; + =

4 А + = 10 ⇒ = 3 + = ⇒ = – = – =

5 С If we turn the second card upside down, we will get the greatest sum + = 16 А One three-leaf clover has leaves; two three-leaf clovers have leaves; three

clovers have 3+3+3=9 leaves; four clovers have 3+3+3+3=12 leaves; five clovers have 3+3+3+3+3=15 leaves

7 А Out of the three expressions only the first is correct

8 В Amina has fish, and together with Sonya they have fish in total If the first two apples are yellow, the third would be red

SPRING 2016: GROUP

Problem

A) B) C)

Problem Which of the following is NOT true?

A) + + = 12 B) + + = 13 C) + + = 14

Problem Which of the following are one-digit numbers? 0, 1, 2, 12, 3, 13 and 14

A) B) C)

Problem

A) B) C)

Problem By how much is 10 greater than 9?

A) 19 B) C)

Problem The sparrows on each tree are as many as the total number of trees The total number of sparrows is How many trees are there?

A) B) C)

Problem I have 17 roses – white, yellow and red The white and yellow roses together are 10, the yellow and red roses together are 10 How many yellow roses are there?

A) B) C)

Problem

A) B) C) 10

Problem I have apple, Yvette has apple more than me, and Daria has apple less than Yvette How many apples in total the three of us have?

Problem 10 Nine children are playing hide and seek One of them is seeking the others and finds of the children How many of the children remain hidden?

A) 16 B) C)

Problem 11 I added two different numbers and the sum I got is Which of the two addends is greater?

Problem 12 Rather than deducting from a certain number, I added and got as a result What is the number I should have gotten?

Problem 13 Three identical balloons cost 10 cents more than a single balloon of the same kind How many cents does a single balloon cost?

Problem 14 One addend is greater than by and the other addend is less than by What is the sum?

Problem 15 I wrote down all numbers from to 16 How many are the digits used more than once?

Problem 16 John needs more balloons in order to have Peter has balloon more than John How many balloons Peter and John have in total?

Problem 17

= + Δ = Δ + =

= ?

Problem 18 There are flowers in a vase, each of which has petals I picked of the flowers What is the total number of petals on the remaining flowers?

Problem 19 A bunny eats carrots every day How many days will it take for the bunny to eat carrots?

Problem 20

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B ⇒

2 C A) true B) true C) NOT true 9+2+4=16

3 B The one-digit numbers are 0, 1, and ⇒ there are four in total 4 B ⇒ ⇒

5 B

6 A If there is tree, there would be sparrow;

If there are trees then the sparrows would be + = 7 B ⏟

+ red = 17 ⇒ red =

yellow + ⏟ = 10 ⇒ yellow = ⇒ There are yellow roses

8 B ⇒

9 A I have Yvette has + = 2, and Daria has – = Together we have + + =

10 C Among the children playing hide and seek, there are hidden children of them have been found

The children that remain hidden are – =

11 2 The addends are different according to the condition of the problem Therefore can be presented as +

The greater of the two addends is

12 0 I added and got as a result Therefore the number that I added to was The result I should have gotten is – =

13 5 Balloon + balloon + balloon = balloon + 10 cents Therefore balloon + balloon = 10 cents

One balloon costs cents

15 3 I wrote down all numbers from to 16: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 The digits which have been used more than once are 1, and They are in total

16 3 John needs balloons in order to have John has balloon Peter has balloon more than John Peter has balloons Peter and John have

1 + = balloons in total

17 6 =1

⏟ ⇒

⏟ ⇒ =6

18 6 There were flowers, each of which had petals I picked of the flowers Now there are flowers in the vase with petals each, i.e the petals are

19 3 A bunny eats carrots every day In two days it would eat carrots In three days it would eat carrots

FINAL 2016: GROUP

Problem

A) B) C)

Problem I have 17 roses – white, yellow and red The white and yellow roses together are 10, the yellow and red roses together are also 10 The roses of which colour are the least in number?

A) white B) yellow C) red

Problem Adam, Bobby, Charles and Daniel won the top four places at a competition Adam was ranked higher than Bobby, Charles was ranked lower than Daniel, and Bobby was ranked higher than Daniel Who came third?

A) Adam B) Bobby C) Daniel

Problem Find the number in the following diagram:

A) B) C)

Problem I will turn 15 in years How old was I years ago?

A) B) C)

Problem How many of the numbers 0, 1, 2, 3, and can be places in the empty square, so that the following equation + < will be true?

A) B) C)

Problem What number will you get as a result of adding the numbers hidden under the shells?

1, 3, , 7, 9, , 13

A) 10 B) 14 C) 16

A) B) C)

Problem There are teams participating in a football tournament Each team played one game against each of the other teams How many games have been played in total?

A) B) C)

Problem 10 There is a basket of apples in a dark room Inside the basket there are yellow and red apples What is the smallest number of apples we would need to take out (without looking), in order to be sure that we have taken out at least two yellow apples?

A) B) C)

Problem 11 The sum of two one-digit numbers is 17 The smaller number was subtracted from the greater number What is the difference?

Problem 12 The numbers 0, 1, 2, and 10 are written down on a piece of paper Annika erased two of the digits and the numbers which remained on the piece of paper were 0, and 10 When she added those numbers she got 11 as a sum Pippi had her own piece of paper which also had the numbers 0, 1, 2, and 10 written on it She also erased two digits, correctly added the remaining numbers, but received a sum smaller than that of Annika What is the smallest possible sum that Pippi could have gotten?

Problem 13 It is well known that when a die is rolled, the winning number is the one found on top of the die (1, 2, 3, 4, or 6)

When the die shown on the picture was rolled, the winning number was Three dice were rolled and there were three different winning numbers The sum of the three numbers was 14 What is the smallest winning number we got?

Problem 14 Find the next (Fifth) sum:

Second sum: + + = Third sum: + + + =

Fourth sum: + + + + =

Problem 15 John arranged 12 books on his shelf The book Pippi Longstocking was arranged 8th from left to right Which place would the same book be at if we were counting from right to left?

Problem 16 The number of one-digit numbers smaller than is less than the number of two-digit numbers smaller than Find

Problem 17 The numbers 1, 2, and must be placed in the following empty squares Find the difference

difference

Problem 18 Arnold and Mary have some pet fish Mary has fish more than Arnold Together they have 18 fish How many fish does Mary have?

Problem 19 All but of a group of 18 children love ice cream How many of the children not love ice cream?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 А ⇒

2 B ⏟

+ red = 17 ⇒ red =

yellow + ⏟ = 10 ⇒ yellow = 3⇒ white =

There are yellow roses

3 С Adam was ranked higher than Bobby ⇒ the ranking is AB Bobby was ranked higher than Daniel ⇒ the ranking is ABD Charles was ranked lower than Daniel ⇒ the ranking is ABDC Daniel came third

4 B + = 13; – =

Therefore 13 - 7= ⇒ =

5 А I will be 15 in years I am currently 15 – = years old, and two years ago I was – = years old

6 В + < 9, correct; + < 9, correct; + < 9, correct; + < 9, correct; + < 9, wrong; + < 9, wrong + < is correct for of the numbers 0, 1, 2, 3, and

7 C The numbers are 1, 3, 5, 7, 9, 11 and 13 The numbers hidden under the shells are and 11 Their sum is 16

8 А Three different sums can be paid using three of the coins: + + = 3; + + = 7; + + = 11

9 А If the teams are A, B and C, then the games played were A and B, B and C, A and C Three in total

10 В In the worst case scenario, we would first take out the two red apples In this case the next two apples would definitely be yellow

12 On the piece of paper that has the numbers 0, 1, 2, and 10 on it, Pippi can erase the following:

10 - two digits and 1; in which case the sum she would get is 10, which is smaller than that of Anika

7 and the first digit of 10; in which case the sum she would get is 3; and the second digit of 10; in which case the sum she would get is and the first digit of 10; in which case the sum she would get is 8; and the second digit of 10; in which case the sum she would get is and the first digit of 10; in which case the sum she would get is 13 The possible options are: 14 = + + = + + = + + = + +

4 The winning numbers are different only in the second sum The smallest winning number was

14 16 The fifth sum is + + + + + = 16

15 After this book, there would be other books if we count from left to right If we count the books from right to left, the book would come after those books, therefore it would come 5th

16 16 The one-digit numbers smaller than are 5: 0, 1, 2, 3, The two-digit numbers smaller than 16 are 6: 10, 11, 12, 13, 14, 15

The number that we would need to place in the square is 16

17 ⏟

⏟

The difference is

18 10 + 10 = 18, therefore Mary has 10 fish

19 Out of 18 children, all but love ice cream Therefore of them not love ice cream

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

Problem There are 14 chocolates in a box Each of the three members of the team ate two chocolates There are

now @ chocolates left in the box Find @

Problem There are # sparrows perched on a bush @ of them flew off the bush The remaining sparrows are less than those who flew off Find #

Problem I have # yellow and red flowers Seven of them are tullips, and the rest are roses Two of the flowers are yellow and the rest are red What is the smallest possible number of red roses? Mark your answer by & Find &

Problem Two identical chocolate bars cost as much as & identical sweets Six chocolate bars cost as much as § sweets Find §

Problem The two-digit numbers smaller than * are § Find *

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ = 2+2+2=6, therefore chocolates have been eaten already There are 14-6=8 chocolates left

2 # = 12 The sparrows left on the bush are 8-4=4 At first the sparrows were 8+4=12

3 & =

The roses are 12-7=5 In order to get the smallest possible number of red roses, both yellow flowers must be roses 5-2=3, therefore at least of the roses are red

4 § =

2 chocolate bars + chocolate bars + chocolate bars are equal to sweets + sweets + sweets = sweets

5 * = 19

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP

Problem What is the missing number?

A) 10 B) 11 C) 21

Problem The sum of 10 + equals:

A) the sum of and 11 B) the difference of 14 and C) the sum of and

Problem In a sum of two numbers, one addend is greater than by 2, while the other addend is smaller than by The sum is:

A) B) C)

Problem What is the largest two-digit number with as a units digit?

A) 10 B) 90 C) 100

Problem How many of the following expressions are correct? 11-2 > 13

18+3 > 20 12-5 = 3+4

A) B) C)

Problem How many are all the possible digits that can be placed instead of @, so that would be true?

A) B) C)

Problem What is the largest sum of different single-digit numbers?

A) 19 B) 18 C) 17

Problem I thought of a number I added it to and got 10 The number I thought of is:

A) 12 B) C) 10

A) 11 B) 19 C) 21

Problem 10 How many are the two-digit numbers that NOT have as a ones digit?

A) B) 81 C) 90

Problem 11 Peter solved problems, Iva solved problems less than Peter; Mary solved one problem more than Iva How many problems did Mary solve?

Problem 12 There is a basket in a dark room In the basket there are yellow and red apples What is the smallest possible number of apples we would need to take out, without looking at their colour, in order to ensure that we have taken out red apples?

Problem 13 How many single-digits numbers is the magic square made of?

6

5

Problem 14 How many sheets of paper are there between the third and the seventh pages of a book?

Problem 15 Find the sum of all two-digit numbers whose sum of digits is 3?

Problem 16 How many numbers have been omitted in the sequence 1, 11, 21, 31, , 81, 91?

Problem 17 Joel has a few bunnies Each one of them has ears and legs If their ears are 10 in total, how many legs they have in total?

Problem 18 If the minuend is and the subtrahend is 9, we get a difference of?

Problem 19 How many units are there in the number equal to

– – – – – ?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B 2 C 10 + = 18, 18 = +

3 C + = 4; – = + =

4 B 90

5 B 9>13; 21>20; 7=7 6 C 36<37; 36<38; 36<39

7 C + = 17

8 B ? + = 10 ? =

9 B 10; 10 – = 9; 10 + = 19

10 B ⏟ ⏟ …., ⏟

9+9+9+9+9+9+9+9+9=81

11 2 Iva solves problem, Maria solved + = problems

12 4 If we were to take both yellow apples, the next would be red Therefore if we take apples, there will always be red apples among them

13 8 6 8 1

0 10

9

14 1 This is the list of paper with page numbers and 15 63 The numbers are 12, 21 and 30 Their sum is 63 16 4 The numbers 41, 51, 61 and 71 have been skipped

17 20 There are 10 ears Therefore the bunnies are Each bunny has legs + + + + = 20

18 0 – =

WINTER 2016: GROUP Problem What is the missing number?

A) B) C)

Problem The sum of is:

A) 90 B) 80 C) 70

Problem In a sum of two numbers, one of the addends is greater than 20 by 20, and the other addend is smaller than 20 by 10 The sum of the two numbers is:

A) 50 B) 40 C) 30

Problem How many of the following expressions are correct?

A) B) C)

Problem What is the missing number „?”?

A) B) 18 C) 35

Problem How many digits can we place instead of @, so that would not be true?

A) 10 B) C)

Problem What is the greatest sum of different one-digit numbers?

A) 23 B) 24 C) 25

Problem There is a basket in a dark room In the basket there are yellow and red apples What is the smallest possible number of apples we would need to take out, without looking at their colour, in order to ensure that we have taken out at least red apples?

A) B) C) 10

Problem If we add the number equal to 94 – (46 + 38) to the number equal to 94 – 46 +38, what result would we get?

A) 86 B) 76 C) 96

Problem 10 A gallery has 96 paintings 32 of them were sold on the first day, and on the second day the gallery sold paintings more than the previous day How many paintings are still not sold?

Problem 11 Three friends weigh respectively 24, 30 and 42 kilograms They want to cross a river by using a boat that can carry a maximum of 70 kg At least how many times would this boat need to cross the river, so that all three of them would get to the opposite shore

Problem 12 How many tens are there in the number equal to

– – – – – ? Problem 13 What is the greatest number in the magic square?

6

2

Problem 14 In how many squares can you find the letter A?

А

Problem 15 Place the digits 1, 2, and in the squares in a way that would result in the greatest sum What is the sum?

Problem 16 Boko and Tsoko went fishing with their sons All of them caught an equal number of fish How much fish did each of them catch, if they caught fish in total?

Problem 17 The minuend is greater than the subtrahend by What is the difference?

Problem 18 How many are the three digit numbers different from 102, that can be derived from the number 102 by randomly moving the digits of the number around?

Problem 19 If we follow the rule:

then which number we need to place in the square with the ant in it?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 A 57 – ? = 56; ? =

2 А 90

3 А One of the addends is 20 + 20 = 40, and the other is 20 – 10 = 10 The sum is 50

4 B 40 – = 38, i.e the first expression is not correct The next two expressions are correct

5 А The missing number in the circle is 35 Then we must add to the number 35, in order to get 43

The number we are looking for is

6 А We need to find out the following: for how many digits @ is it NOT true that: 40 > 4@?

For all ten digits: 0, 1, , 7 В + + = 24

8 B In the worst case scenario, we would take out all of the yellow apples first Then after more attempts, we would have taken out red apples, i.e in total

9 С The first addend is 10, and the second is 86 The sum is 96

10 С The paintings sold on the second day were 35 The paintings sold on the first and second day together are 67

The paintings that remain unsold are 96 – 67 = 29

11 3 Let C denotes the heaviest of the three friends, A - the lightest one, and B - the third one

It would be impossible for all three of them to cross the river in one go, because 24 + 30 + 42 = 96 > 70

Therefore the boat would have to return at least once, and the smallest possible number of river crossings would be

Following is an example of a way in which all three friends can cross the river to the opposite shore:

A crosses back to the initial shore

A and C now cross to the opposite shore together

12 10 – – – – – = 20 + 20 + 40 + 20 + = 100 In the number 100 there are 10 tens 13 10 The magical sum is 15

The numbers in the second row are 0, and 10, and in the third row are 9, and

The greatest number is 10

14 4 The letter A is in one square 1, in two squares 2 and in one square 3 15 46 + + 43 = 46

16 3 or If we assume that the problem speaks of four people – two fathers and two sons, then the result would be impossible, because is not divisible by Therefore the problem must speak of three people: a grandfather, his son, and his grandson, or of people: two fathers and seven sons

17 2  + –  =

18 3 The numbers are 102, 120, 201 and 210 One of them has been written down already 19 0 The numbers are as follows:

At the bottom: 9, 5, 2, Above: 4, 3,

Above: 1,

And the number at the top is

SPRING 2016: GROUP

Problem If –  then  =

A) 100 B) 99 C) 98

Problem Which of the following lengths is the shortest?

A) mm B) cm C) dm

Problem If  , then  =

A) B) C)

Problem I chose a random number I switched the numbers of ones and tens I added 19 to the resulting number and got 24 What is the number I had originally chosen?

A) 15 B) 50 C) 51

Problem Alia and Daniel had 24 sweets at first Then Alia bought more sweets and she now has 12 sweets more than Daniel How many sweets does she have at the moment?

A) 18 B) 19 C) 20

Problem The even numbers from to , inclusive, is 20 What is the greatest possible value of ?

A) 41 B) 42 C) 43

Problem Which of the following numbers is the smallest?

A) + 2 B) 13 – C) (3 + 2)

Problem The number of sparrows on each tree is equal to the total number of trees The total number of sparrows is 16 How many trees are there?

A) B) C)

Problem Two two-digit numbers have been written using different digits Which of the following sums is possible?

A) 22 B) 26 C) 33

Problem 10 I bought stamps, worth cents each, and I payed using coins of 10 cents In how many different ways can I get my change?

Problem 11 The numbers 1, 2, 3, and are written down on two pieces of paper The product of the numbers from one of the pieces is equal to the product of the numbers from the other piece How many numbers are there on the piece of paper that has the number 1?

Problem 12 There are grandmothers, mothers, daughters and granddaughters in a room What's the smallest possible number of people in that room?

Problem 13 There are 22 students in a class Twelve of the students have the highest grade in less than four subjects, and 12 have the highest grade in more than two subjects How many students have the highest grade in exactly three subjects?

Problem 14 In Rose’s garden there are 88 roses which are not in bloom yet and which are blooming Every day new roses bloom and the ones that are already blooming not fade How many days will it take for the blossoming and non-blossoming roses to be an equal number?

Problem 15 Replace the smileys with two of the cards in order to get the greatest possible product

What is the greatest possible product?

Problem 16 The square is ‘magical’ Calculate the number A

21 18 27 15 А

24

Problem 17 If ⏟

⏟ 

, then  =

Problem 18 The product of five numbers is What is their sum?

Problem 19 A container full of water weighs 20 kg and when half full it weighs as much as empty containers How many kilograms does this container weigh when it is empty?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B 100 –(29 + 37) =  – 65, then 100 – 66 =  - 65 34 =  - 65  = 99

2 A A) mm B) cm = 20 mm C) dm = 10 cm = 100 mm 3 C If  , then 

17 = 25   =

4 B The number with exchanged digits of the ones and tens is 24 – 19 = Therefore the originally chosen number is 50 From 50 we can get 05 = and + 19 = 24

5 B Before buying the extra sweets, Alia had 10 sweets more than Daniel 24 – 10 = 14 and 14 = 7, therefore before buying the extra sweets Alia had 17 sweets and Daniel had At the moment Alia has 19 sweets

6 C The 20 even numbers from onwards are 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42

The even numbers from to 42, inclusive, are 20 The even numbers from to 43, inclusive, are 20 The even numbers from to 44, inclusive, are 21

7 A + 2 = + = 7; 13 – = 13 – = 10; (3 + 2) = 10 8 B If the trees are 3, the sparrows would be 3 = 9;

If the trees are 4, the sparrows would be 4 = 16; If the trees are 5, the sparrows would be 5 = 25

10 C The change would be equal to 10 – = cents It can be given in different ways:

5 + = + + = + + + 1= + + + + = + + 1+ +1+1

11 3 The product of the numbers is equal to = 144 Therefore we would need to write numbers that have a product of 12 on the pieces of paper

The numbers can be written down as follows: 1, and on the first piece of paper, and on the second piece of paper, or and on the first piece of paper, 1, and on the second piece of paper The pieces of paper that has the number on it has numbers written on it

12 6 In order for one of the women to be a grandmother, she would need to have a daughter, and a granddaughter Therefore if there are two

grandmothers, who are also mothers, they have one daughter each, i.e daughters, each of whom is also a mother to granddaughter –

granddaughters, who are also daughters The two granddaughters are also daughters

There are now daughters left, who are also mothers There are now mothers left, who are also grandmothers

13 2 The total number of students in the class plus the number of students who have the highest grade in subjects equals 12 + 12 = 24 If we calculate 24 - 22 we would get the number of students who have the highest grade in subjects, i.e

14 10 The roses in blossom and those not yet in blossom are 96 in total The number of roses in blossom must increase by 96 – = 40 roses That can happen in 40 = 10 days

15 63 The possible products are 6; 7; 7; 9; The greatest among them is 63

16 3 We can find the answer by comparing the sums of the numbers from the first column (B, 27, C) to the diagonal (B, 15, 24)

They are equal, B + 27 + C = B + 15 + 24, therefore 27 + C = 39 We get that C = 12, therefore the ‘magical’ sum is 45 (12 + 15 + 18) 27 + 15 + A = 45, therefore A =

17 5 If ⏟

⏟



,

then 20 =   = 18 9 = 1 1,

therefore the sum we are looking for is + + + + =

19 4 The weight of the water in a half-full vessel is equal to two empty vessels The weight of the water in a full vessel weighs as much as empty

vessels The weight of the vessel plus the water inside it is equal to empty vessels

Therefore one empty vessel would be equal to 20 = kg

20 2 A B C D

A + + +

B + – +

C + – –

D + + –

If we add the number of hand shakes, the number must be divisible by 2, because each hand shake is counted twice

In this case the number of hand shakes is + x

We can mark the number of David’s handshakes with x The number x can NOT be greater than

6 + x can be divided by only if x is either or

FINAL 2016: GROUP

Problem The product of all even one-digit numbers that are divisible by 3, is:

A) B) C) 18

Problem What number should be placed instead of  so that the following equation would be true? 

A) 35 B) 36 C) 37

Problem When Adam was counting the numbers from to 50, he got distracted and he forgot to count the numbers that are divisible by or by How many numbers, smaller than 31, did he forget to count?

A) 20 B) 10 C)

Problem There are 20 odd numbers from to , inclusive What is the greatest possible value of ?

A) 41 B) 42 C) 43

Problem Adam, Bobby, Charles and Daniel won the top four places at a competition Adam was ranked higher than Bobby, Charles was ranked lower than Daniel, and Bobby was ranked higher than Daniel Who came third?

A) Adam B) Bobby C) Daniel

Problem A container full of water weighs 21 kg and when half full it weighs as much as empty containers How many kg of water are there in the container when it is full?

A) B) 16 C) 18

Problem When I grow years older than I am now, I will be twice as old as my brother who was born years ago How old am I at the moment?

A) 10 B) 12 C) 20

Problem How many numbers can we place in the empty square, so that the following equation would be true?

 < 25?

A) B) C) more than

Problem By how much is the number hidden under the first shell smaller than the number hidden under the second shell?

4, 7, 13, , 34, , 67

Problem 10 We are given the numbers 1, 2, and If we erase two of them, then the product of the remaining numbers can be presented as the product of two equal multipliers Which numbers should we erase to that?

A) and B) and C) and

Problem 11 A few football teams are participating in a tournament After a game has been played, only the winner moves forward into the tournament If the teams are 16, what is the minimum number of games that must be played in order for one of the teams to become a champion?

Problem 12 There is a basket of apples in a dark room There are yellow and red apples inside it What is the minimum number of apples you would need to take out (without looking) in order to be sure that you have taken out yellow and red apples?

Problem 13 The sum of 11 one-digit numbers is 98 What is the smallest among these numbers?

Problem 14 On the figure below you can see that in the middle there is a square with a side of 1cm On each of its sides there is another square, each with sides of 1cm On each of the sides of the newly formed figure, there is one extra square with a side of 1cm How many squares are there in total on the figure?

Problem 15 Here is what a few children said about the number 63: Adam: “This is a number made up of odd numbers!”

Bryan: “This number is a product of the numbers and 9!” Steve: “This number has 63 units!”

Problem 16 I bought sweets, each of which costs cents, and I paid using coins of 10 cents In how many different ways can the shopkeeper give me my change?

Problem 17 There are 26 students in a class of second-graders 15 of them have less than four balloons, and 17 have more than two balloons How many of the students have more than three balloons?

Problem 18 What is the smallest possible sum of the numbers that we would need to place in the empty squares, so that the sum of the numbers in order of rows, diagonals, and columns would be the same?

2 2

Problem 19 The digits used to write down the even two-digit numbers are more than the digits used to write down the odd one-digit numbers By how many?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 А The even one-digit numbers divisible by are and Their product is

2 А We can write down the equality as follows:

3 + 15 + 35 + 63 = + 16 + 25 + 36 +   = 35

3 А

He forgot to count all even numbers, of which there are 15, as well as all odd numbers divisible by 3, which are 3, 9, 15, 21 and 27

He forgot to count 20 numbers in total

4 В

The 20 odd numbers from onwards are 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,41

The odd numbers from to 41, inclusive, are 20 The even numbers from to 42, inclusive, are 20

5 C The four boys are ranked as follows: AB and DC, therefore ABDC

6 С

If the half-full container weighs as much as empty containers, then the weight of the water in a half-full container is equal to the weigh of empty containers The weight of the water in a full container is equal to the weight of empty containers I.e when full of water, the container weighs as much as empty containers Therefore one empty container weighs The water in a full container weighs 21 – = 18 kg

7 В

We can find the correct answer by checking each possible answer If I am 10 years old now, then my brother is years old In years I will be 18 and my brother will be 10 The number 18 is not twice as big as 10 If I am 12 now, my brother is In years I will be 20 and he will be 10 This is the correct answer

8 В All numbers from to (5 numbers)

9 С

4, 7, 13, , 34, , 67

7 = + 3; 13 = + 3;

The next number is 34 + = 49, The next number is 49 + = 67

The difference we are looking for is 49 – 22 = 27

10 В If we were to erase the numbers and 3, we would get a product of 4, which we can present as

11 15

First we split the 16 teams into couples

They play games, therefore there are winners teams carry on to the second round

8 teams play games in the second round

4 teams carry on to the third round, to play games Final: teams play game

The games played in total are + + + = 15

12 In the worst case scenario we would take out all yellow apples first, and the 5th apple would be red

13 The sum of 11 one-digit numbers can be 99 at most In this case it is 98 Therefore, one of them is

14 18 There are 14 squares with a side of cm on the figure; squares with a side of cm and square with a side of cm There are 18 squares in total 15 Only Adam’s claim is not true

16

The change is 70-63=7 cents

I can get my change in different ways: coins of cent;

5 coins of cent + coin of cents; coins of cent + coins of cents; coins of cent + coin of cents; coin of cent + coins of cents; coin of cents + coin of cents

17 11 15+17-26=6 children have balloons each 17-6=11 students have more than balloons each

18

1 0 1 2

We can compare the sums of the numbers from the first row and from the first column They are equal, therefore they must have two more equal numbers each - х

2 x x 2

We can then compare the second row and the second column: x

x 2 x

We can then compare the first row with the second column and the first row with the third column and we would get the following:

y x x y 2 x y

If we then compare the sums along the diagonals, we will get that

y + у = + x the smallest possible value is y=1 and x=0

The sum we are looking for is

19 85 or There are 45 even two-digit numbers and they have been written down using 90 digits The odd one-digit numbers are written down using digits The answer we are looking for is 90 – = 85

Another answer is also possible:

The even two-digit numbers are written down using 10 digits and the odd one-digit numbers are written down using digits In this case the answer would be 10 – =

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem The number of two-digit numbers that can be presented as a product of two consecutive numbers is @ Find @

Problem If the dividend is ⏟

, and the divisor is 7, what is the quotient #?

Problem Little Red Riding Hood needs to cross a river by going through the only bridge, in order to get to her grandmother’s village She can reach the bridge using & different roads, and she can use two different roads from the bridge to her grandmother’s village It turns out she can reach her grandmother’s village using # different routes Find &

Problem Bugs Bunny loves eating cabbage and carrots He eats either &+1 carrots or cabbages every day In one week Bugs Bunny ate 30 carrots and § cabbages Find §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ =

2 = < 10;

= 12; 5= 20; = 30; = 42; = 56; = 72; 10 = 90;

10 11= 110 >99

The number we are looking for is @ =7 2 # = The dividend is + + + + 10 + 12 + 14

The quotient is 56 7=8 3 & = & = # = & =

4 § =4

Bugs Bunny eats carrots a day It would take him days to eat 30 carrots He would only eat cabbage on the seventh day – he would have to eat cabbages

§=4

5 * =

We found that §=4

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP

Problem In how many ways can we place a digit instead of @ so that will be true?

A) B) C)

Problem The unknown addend @ in the equation dm = @ dm + 20 cm is:

A) B) 10 C) 17

Problem To what number you add 13 to get 101?

A) 87 B) 88 C) 89

Problem The number is NOT the sum of:

A) consecutive numbers B) consecutive numbers C) consecutive numbers Problem How many even numbers is the magic square made of?

6

5

Hint: is an even number

A) B) C)

Problem How many sheets of paper are there between the 23 and 77 pages of a book?

A) 25 B) 26 C) 55

Problem From which number, if we subtract 11, we will get 89?

A) 80 B) 90 C) 100

A) 11 B) 21 C) 29 Задача Fill in the missing number in the box

 + 15  25 =

A) 18 B) 15 C) 12

Problem 10 How many are the numbers between 12 and 120, which have at least two digits of 1?

A) 10 B) 11 C) 12

Problem 11 What is the number of possible different sums that we get when we add the results from throwing dice?

Problem 12 Find the value of

Problem 13 There were pieces of paper Some of them were cut into three parts Altogether, there are now 19 pieces of paper How many pieces were cut into three parts?

Problem 14 A textbook is opened at random To what pages is it opened if the sum of the facing pages is 89?

Problem 15 What are the last digits of the sum

⏟

Problem 16 How many numbers between and 99 are divisible by and 6? Problem 17 It is known that :

- Among A, B , C and D there are two excellent students; - Among A, B and C there is one excellent student; - Among A, C and D there is one excellent student How many are the excellent students?

Problem 18 How many seconds we have to take out of 72 seconds to get minute?

Problem 19 Use 1, 2, 3, and to form a 2-digit number and a 3-digit number Find the largest sum of these two numbers

Problem 20 How many sticks with a length of 11 cm can we cut off from a stick with a length of

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B) 139>100; 139>110; 139>120; 139>130

2 A) 3 dm = dm + 20 cm

3 B) 88 101 – 13 = 88

4 B) 3=1+2; 3=0+1+2

5 A) 6 8 1

0 10

9

6 B) 26 These are the lists of paper with page numbers (25,26),…, (75,76)

7 C) 100 100 – 11 = 89

8 B) 21 If the first 20 pencils are two of the colors, the 21st will be of the third color

9 B) 15 ((0+25) - 15)÷ = 15

10 B) 11 101,110, 111, …, 119

11 21 The smallest sum is 1+1+1+1=4, …, the greatest is 6+6+6+6=24 From to 24 the possible sums are 21

13 5 If we cut list of paper into smaller lists, their number would be + = 11

If we cut lists of paper into smaller lists, their number would be + = 13

If we cut lists of paper into smaller lists, their number would be + = 15

If we cut lists of paper into smaller lists, their number would be + 12 = 17

If we cut lists of paper into smaller lists, their number would be + 15 = 19

14 44 and 45 89 = 44 + 45

15 85 1+4+9+16+25+36+49+64+81=285

16 16 The numbers are 6, 12, 18, …, 90, 96

17 2 If A is an excellent student, then from the second and third statement it follows that B, C and D cannot be excellent students Therefore of the statements is not true A is not an excellent student

If B is an excellent student, then C cannot be an excellent student (as follows from the second statement) Therefore D must be an excellent student

If B is not an excellent student, then C and D would be excellent students (first statement) However in this case the third statement could not be true

Answer: The excellent students are B and D There are two of them 18 12 72 – 12 = 60 seconds = minute

19 573 573= 542+31=541+32=531+42=532+41

WINTER 2016: GROUP

Problem What is the missing number?

A) 142 B) 258 C) 242

Problem The sum of is:

A) 750 B) 730 C) 650

Problem What is the missing number?

A) B) C)

Problem How many of the following expressions are correct?

A) B) C)

Problem

A) B) C)

Problem What is the sum of the numbers in the 9th row?

1

4

7

A) 99 B) 78 C) other

Problem There is a basket in a dark room In the basket there are yellow, red and green apples What is the smallest possible number of apples we would need to take out, without looking at their colour, in order to ensure that we have taken out apples from all three colours?

Problem If a person participating in an archery contest hits the target in all attempts, how many different results can they get? Keep in mind that the result is the sum of all attempts (If the arrow falls between and 9, points are counted If it falls on the line between and 10, 10 points are counted.)

A) B) C)

Problem If we add the number equal to 300 + 100 to the number equal to 200 + 400, we would get

A) 100 B) 1,000 C) 1,100

Problem 10 A gallery has 360 paintings 101 of them were sold on the first day The paintings sold on the second day were 31 more than those sold on the first day How many paintings are still not sold?

А) 132 B) 127 C) 228

Problem 11 Three friends weigh respectively 24, 30 and 42 kilograms They want to cross a river by using a boat that can carry a maximum of 70 kg At least how many times would this boat need to cross the river, so that all three of them would get to the opposite shore?

Problem 12 By how much is the larger sum greater than the smaller sum?

Problem 13 In how many rectangles can we find the ant? (Keep in mind that a square is also a rectangle.)

Problem 14 Place the digits 1, 2, 7, and in the squares in such a way that after calculating, the result would be the greatest possible number What is the number?

+ +

Problem 15 Boko and Tsoko went fishing with their sons All of them caught an equal number of fish How much fish did each of them catch, if they caught fish in total?

Problem 16 What is the greatest possible sum of three odd one-digit numbers?

Problem 17 The numbers 1, 10, 19, 28,…, 82, 91 have been written down according to the following rule: we get each following number by adding to the preceding number, until we reach 91 How many numbers have been written down?

(Hint: 10 = + 1 9; 19 = + 9; 28 = + 9, … )

Problem 18 I chose a number I subtracted 555 from it and got 166 as a result What was the number I chose?

Problem 19 What is the smallest three-digit number with 18 as the sum of its digits?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 В ? – 58 = 200; ? = 200 + 58 = 258

2 А 247 + 178 + 325 = 425 + 325 = 750

3 А = 30;

=

4 С All three expressions are correct 5 С The value of the expression is

The possible answers are respectively 24, 24 and

6 В The numbers in the 9th row are 25, 26, and 27 (read from left to right) Their sum is 78

7 B In the worst case scenario we would first take out all yellow apples, after that we would take out all red apples, after which we would take out green apple We would then have taken out apples from all three colours

6 + + = 12

8 С The smallest result is 24, and the greatest is 30 There are possible results in total:

24 = + + 8; 25 = + + 9; 26 = + + = + + 10; 27 = + + = + + 10; 28 = + + 10 = + 10 + 10; 29 = + 10 + 10; 30 = 10 + 10 + 10 9 B (300 + 100) + (200 + 400) = 400 + 600 = 1000

10 В The paintings sold on the second day were 101 + 31 = 132 The paintings sold on the first and second day together were 233 The paintings remaining unsold in the gallery are 360 – 233 = 127

11 3 Let C denotes the heaviest of the three friends, A - the lightest one, and B - the third one

It would be impossible for all three of them to cross the river in one go, because 24 + 30 + 42 = 96 > 70

Therefore the boat would have to return at least once, and the smallest possible number of river crossings would be

to the opposite shore:

C stays on one of the shores, while A and B cross over to the opposite shore A crosses back to the initial shore

A and C now cross to the opposite shore together

12 219

The first sum is 136, and the second sum is 355 355 – 136 = 219

13 6 The ant can be found in square 1, in rectangles 2, in rectangle 3; in square 2, in rectangle

14 12 + + – 12 = 12

15 3 or If we assume that the problem speaks of four people – two fathers and two sons, then the result would be impossible, because is not divisible by Therefore the problem must speak of three people: a grandfather, his son, and his grandson, or of people: two fathers and seven sons

16 27 + + = 27

17 11 We get the second number by adding once (1 + 9) to We would get 91 by adding the number ten times to the number (1 + 10 9) Therefore the numbers that have been written down are 11

18 721 555 + 166 = 721

19 189 The number is presented as 1xy, where x + y = 17 17 is presented in two ways: as the sum of the two numbers and 8, or and The numbers of the type 1xy are two: 189 and 198 The number we are looking for is 189

20 50 + 91 + 18 + 82 + 27 + 73 + 36 + 64 + 45 + 55 = (9 + 91) + (18 + 82) + (27 + 73) + (36 + 64) + (45 + 55) = 100 + 100 + 100 + 100 + 100 = 500

SPRING 2016: GROUP

Problem If then = ?

A) 54 B) 48 C) 60

Problem 1,000 – (12 + 23 + 34 + 45 + 55 + 66 + 77 + 88) = ?

A) 400 B) 500 C) other

Problem One kg of dried mushrooms is derived from 12 kg of fresh mushrooms How many kg of fresh mushrooms would you need to get kg of dried mushrooms?

A) B) 18 C) 72

Problem Two ants are moving towards each other One of them travelled a distance of 176 cm, and the other travelled 80 mm more than the first What is the length that both ants travelled in total?

A) 36 dm B) 260 cm C) 402 mm

Problem The product of natural numbers is 72 The sum of these numbers is 15 and neither of them is Which is the greatest among these numbers?

A) B) C)

Problem We have identical chocolate bars, each consisting of 28 pieces We have to divide them equally between children What is the minimum number of times we need to break each chocolate bar in order to this?

A) B) C)

Problem The sum of the three-digit numbers ̅̅̅̅̅, ̅̅̅̅̅̅ and ̅̅̅̅̅ is 1010 ( and represent missing numbers) In this case, what is the three-digit number ̅̅̅̅̅̅

A) 382 B) 371 C) 473

Problem A book has been numbered as follows: the first pair of pages has been numbered as and 2; the second pair as and 4, and so on, until the last pair of pages, which has been numbered as 127 and 128 If I open the book at a random place, what is a possible product of the numbers of the two pages that I’ve opened the book at?

A) 90 B) 72 C) 56

A) B) C)

Problem 10 How many digits are used to write down the first 100 odd numbers?

A) 250 B) 245 C) 200

Problem 11 Four children met together: Adam, Bobby, Charley and Daniel Adam shook hands with of these children, Bobby shook hands with 2, and Charley shook hands with How many of the children’s hands did David shake?

Problem 12 I solve problems a day, and my brother solves three times less Together we solved 72 problems How many days did it take us to this?

Problem 13 The product of a few different one-digit numbers is a number that has a in the ones place How many even numbers are there among the multipliers?

Problem 14 Between each two neighbouring digits of the number 2016, I placed either addition signs and multiplication sign, or multiplication signs and addition sign

Example: or

How many different numbers will I get after calculating all such expressions?

Problem 15 Annie has a magical necklace Each bead of the necklace is numbered (1, 2, 3, and so on) If between the beads numbered as and 15 there is the same number of beads, what is the total number of beads on Annie’s necklace?

Problem 16 In Rose’s garden there are 232 roses which are not in bloom yet and 168 which are blooming Every day new roses bloom and the ones that are already blooming not fade How many days will it take for the blossoming and non-blossoming roses to be an equal number?

Problem 17 A vessel, when full of water, weighs 20 kg, and when half full it weighs as much as empty vessels How many kilograms does this vessel weigh when it is empty?

Problem 18 A square has a side length of cm On each of its sides (on the outside) has been built another square with a side length of cm After that, on each of the sides of the new figure, another square with a side length of cm has been built How many cm is the perimeter of the final figure? Problem 19 What is the maximum possible number of different odd three-digit numbers that we can add and receive a three-digit number as a result?

Problem 20 The expression we are going to use for this problem is

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 A ⇒ =54 2 C 1000 – (12 + 23 + 34 + 45 + 55 + 66 + 77 + 88) =

1000 – (100 + 100 + 100 + 100) = 1000 – 400 = 600

3 C One kg of dried mushrooms can be derived from 12 kg of fresh mushrooms In order to get kg of dried mushrooms we would need 12 = 72 kg fresh mushrooms

4 A One of the ants travelled a distance of 176 cm, and the other traveled 176 + = 184 cm The distance that both ants travelled in total is equal to 176 + 184 = 360 cm = 36 dm

5 В 15 = + + + = + + + = + + + Therefore the number we are looking for is

6 А The number of all pieces of all five chocolates is 28 = 140 Therefore each child should get 140 = 20 pieces

By breaking one chocolate, we can get 20 pieces and have left In this way we can give 20 pieces to children, however there would be more children and more parts, each consisting of pieces, left

We can give parts with pieces each to each of the two children, and the fifth part, which consists of pieces, we can divide in parts of pieces The number of times we would need to break the chocolates is + = 7 B ̅̅̅̅̅ + ̅̅̅̅̅̅ + ̅̅̅̅̅ = 1010

⇒ ⇒

⇒ Therefore ̅̅̅̅̅̅

8 B If I open the book, there will be two pages, both numbered The smaller number will be even, and the greater will be odd The numbers will be consecutive

90 = 10; 72 = and 56 = 8, therefore I have opened the book at the pages numbered and

grandmothers, who are also mothers, they have one daughter each, i.e daughters, each of whom is also a mother to granddaughter – granddaughters, who are also daughters

The two granddaughters are also daughters

There are now daughers left, who are also mothers There are now mothers left, who are also grandmothers

10 B Among the first 100 odd numbers there are one-digit numbers, 45 two-digit numbers and 50 three-two-digit numbers

Therefore the number of digits which have been used to write them down is + 45 + 50 = 245

11 2 A B C D

A + + +

B + – +

C + – –

D + + –

If we add the number of hand shakes, the number must be divisible by 2, because each hand shake is counted twice

In this case the number of hand shakes is + x

We can mark the number of David’s handshakes with x The number x can NOT be greater than

6 + x can be divided by only if x is either or

However, x is not 0, because Adam shook hands with all the children Therefore x = David shook hands with children

12 9 I solve problems a day, and my brother solves Together we solved problems in total It would take us days to solve 72 problems

13 0 The number is among the multipliers If there is at least one even number, then the product would be divisible both by 2, and by 5, i.e by 10, it would have a ones digit of Among the multipliers there are no even numbers

14 4 Here are all the different options:

The result consists of numbers: 2, 6, and

15 20 The beads with numbers from to 14 are situated between the beads with numbers from to 15 The beads are in total The beads on the opposite side are also If we also note the beads numbered and 15 we will find that the beads on Annie’s necklace are + = 20

16 8 The roses which are blooming and the roses which are not yet in bloom are 400 in total The number of the roses in bloom needs to be increased by 32 roses This will happen in 32 = days

17 4 The weight of the water in a half-full vessel is equal to two empty vessels The weight of the water in a full vessel weighs as much as empty vessels The weight of the vessel plus the water inside it is equal to empty vessels Therefore one empty vessel would be equal to 20 5=4 kg 18 20 There are squares with sides each and squares with sides each

which form the final figure Therefore we get that + = 20 cm 19 9 101 + 103 + 105 + 107 + 109 + 111 + 113 + 115 + 117 = 981;

101 + 103 + 105 + 107 + 109 + 111 + 113 + 115 + 117 + 119 = 1100 20 2 In order for the value of the expression to be increased by 1, the following

needs to be true:

The first number in the expression 3+12-10 needs to be exchanged with Then the initial value would be increased by

2= would be possible if we exchange for = is not possible

The second number in the expression 3+12-10 needs to be exchanged with 13 Then the initial value of the expression would be increased by

3= 13 is not possible =13 is also not possible

The third number in the expression 3+12-10 needs to be exchanged with Then the initial value would be increased by

1 = 9, if we exchange 10 for 10=9 is not possible

FINAL 2016: GROUP

Problem If ̅̅̅̅̅ then Δ = ?

A) 10 B) 12 C) 14

Problem 1000 – (5 + 15 + 25 + 35 + 45 + 55 + 65 + 75 + 85 + 95) = ?

A) 400 B) 500 C) 600

Problem If we have kg of fresh mushrooms and we dry them, we would get 100 g of dried mushrooms How much fresh mushrooms we need in order to derive kg of dried mushrooms? A) 10 kg fresh mushrooms B) 20 kg fresh mushrooms C) 30 kg fresh mushrooms

Problem The segment AB is km long and has been divided into 1000 equal parts by a number of points The points have been numbered, with A being the first point and B being the last point Point C is found at an equal distance between point 101 and point 203 What is the distance (in meters) from point A to point C?

A) 150 B) 151 C) 152

Problem Iva arrived at the bus stop and looked at her watch, which showed the time to be 08:01h, which meant that she was minutes late for her bus What she did not know was that her watch was running minutes ahead If the bus came minute late, how many minutes did Iva have to wait at the bus stop?

A) B) C) more than

Problem Now many times at least would we need to break chocolates, in order to divide them equally between children? Each chocolate is made up of 28 pieces

A) B) C)

Problem A book is numbered as follows: the pages on the first sheet are numbered as and 2, the pages on the second sheet are numbered as and 4, and so on, until the last sheet, where the pages have been numbered as 47 and 48

If I were to rip off consecutive sheets and then add the numbers with which the pages of the sheets have been numbered, which of the following sums would I get?

Problem If we exchange the identical letters with identical numbers, and the different letters with different numbers, then what would the greatest possible value of the following expression be equal to?

A) 157 B) 156 C) 158

Problem Adam wrote down 35 numbers The first number he wrote is 7, and each next number is twice as big as the preceding one How many of the numbers he wrote down are greater than 224?

A) 30 B) 29 C) 28

Problem 10 On the figure below you can see that in the middle there is a square with a side of 1cm On each of its sides there is another square, each with sides of 1cm On each of the sides of the newly formed figure, there is one extra square with a side of 1cm What is the minimum number of squares that must be erased in order for only 15 squares to remain on the figure?

A) B) C) more than

Problem 11 If the first day of the year is a Monday, what would the last day of the same year be? Problem 12 Alex and Boris each have coins of 1, and cents Boris used of those coins to add up the smallest possible sum and Alex used of those coins to add up the greatest possible sum By how much is Boris’ sum smaller than that of Alex?

Problem 14 You are given the following expression: – Exchange one of the numbers in the expression with a different number, so that the initial value of the expression would be increased by In how many ways can we this?

Problem 15 How many times is the number hidden under the first shell smaller than the number hidden under the second shell?

1, , 2, 6, 24, , 720, 040

Problem 16 At least how many of the numbers we need to change in order for the product of the numbers along the diagonals, rows and columns to be the same?

1

16

2

Problem 17 Each of the 10 digits has been used once to write down two-digit numbers with the greatest possible sum What is the sum?

Problem 18 At a football game, the winner earns points and the loser earns points If the match is drawn, both teams get point each After having played games, a team had earned 11 points What is the possible number of losses that the team had?

Problem 19 A number is perfect when the sum of its divisors (except the number itself) equals the given number The number is called perfect because it is equal to the sum of

1 + + 3, where 1, and are all its divisors, except the number The next perfect number is an even number greater than 24 and smaller than 30 What is the number?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 В 332 4=1328 ⇒ Δ=12

2 B

1000 – (5 + 15 + 25 + 35 + 45 + 55 + 65 + 75 + 85 + 95) = = 1000 – 500 = 500

3 B

2 kg = 2000 grams = 20 100 grams, which means that we would need 20 kg of mushrooms

4 В

Point C, which is found at an equal distance between point 101 and point 203, is point 152 The distance from the first point to the 152nd point is 151 meters

5 А

The watch shows that the time is 8:01h She is minutes late, which means that she was supposed to arrive at 7:59h, according to her watch This means that she came minutes early, because the bus (according to her watch) was supposed to arrive at 8:04h The bus is running minute late Therefore it would arrive at 8:05h Iva had to wait for minutes

6 А

Each child will receive chocolate plus extra 14 pieces of the remaining chocolates In this case we would only need to break the chocolates twice

7 C

Everything follows from: ⏟

⏟

⏟

45

9 В

The numbers are 7, 14, 28, 56, 112, 224, 448, of them are smaller than 224, is equal to 224, 35 – = 29 are greater than 224

10 А

The 1 squares are 13, the 2 squares are 4, and there is one 3 square There are 18 squares in total

If we were to remove one 1 square, the 1 squares would be 12, the 2 squares would be 3, and the 3 squares would be 0, therefore the total number of squares would be 15

This can be done in ways, by removing square from one of the angles of the the 3 square

11

Monday or Tuesday

The calendar year has 365 days, or 366 days on a leap year When dividing 364 and 365 by 7, the remainders are and Therefore the last day of the year would either be a Monday or a Tuesday

12 8

The smallest possible sum is + + = 14, and the greatest possible sum is + + 1 = 22 The difference is 22 – 14 =

13 А or D

First we need to arrange people as follows: DCEA or AECD In this case B may be situated as follows:

ВDСЕA DСЕAВ АECDВ ВАECD

14 3

– ; – ; – We can that in ways

15 120

The rule is as follows: multiply the first number by 1, and get the second; multiply the second number by 2, and get the third, etc

1; 1 = 1; = 2; = 6; = 24, 24 = 120; 120 = 720;

The number hidden under the first shell is 1, and the number hidden under the second shell is 120 120 = 120

16 1

If we exchange the number in the first row, first column, with a 2, we would get a magical square

2 16 4

17 360 90+81+72+63+54=360

18 0 or

If a team were to win all games, they would earn 21 points This team however earned 11 points, which means that they lost 10 points from the maximum score

This can be done through: drawn games and losses, drawn games and losses; Answer: or losses

19 28 The number is 28 28 = + + + + 14

20 450

TEAM COMPETITION – NESSEBAR, BULGARIA MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem If ⏟

a n s , find @

Problem Our rabbit now has less than @ male and female bunnies Each male bunny has as many sisters as brothers, and each female bunny has half as many sisters as brothers If the number of bunnies our rabbit has is #, then find #

Problem Find the smallest even three-digit number &, if it is known that & is divisible by #

Problem The number #+2 is presented as the product of consecutive odd numbers with a sum of § Find §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ = 10

⏟

40 ⇒@ = 10

⏟

a n s

40 ⇒@ = 10

2 # =7

The condition says that each male bunny has as many sisters, as brothers Therefore the number of baby bunnies our rabbit has is 3, 5, or We can check each possible answer:

If the correct answer is 3, then each male bunny would have brother and sister However the only female bunny in this case would not have “half as many sisters as brothers”

If the correct answer is 5, then each male bunny would have brothers and sisters However in this case each female bunny would have sister and brothers, therefore it would not have not have “half as many sisters as brothers”

If the correct answer is 7, then each male bunny would have brothers and sisters In this case each female bunny would have sisters and brothers, which means that the condition is satisfied – each female bunny would have “half as many sisters as brothers”

If the correct number is 9, then each male bunny would have brothers and sisters In this case each female bunny would have sisters and brothers, therefore it would not have “half as many sisters as brothers”

3 & =103

If we carry out a check, we will find that among the differences 100 5, 101 5, 102 5, 103 5, 104 5, 105 5, , the first one that is divisible by #, i.e by 7, is 103 The number we are looking for is 103

& = 103

4 § =16 The number 103 + = 105 = The sum of these multipliers is 16

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP Problem What is the largest 4-digit number with as a units digit?

A) 9,909 B) 9,990 C) 9,099

Problem How many are all the possible digits that can be placed instead of @, so that would be true?

A) B) C)

Problem I thought of a number I added it to 222 and got 1,000 The number I thought of is:

A) 1,222 B) 888 C) 778

Problem How many are the 5-digit numbers that NOT have as a units digit?

A) 810 B) 8,100 C) 81,000

Problem How many sheets of paper are there between the and the pages of a book?

A) 99 B) 98 C) 48

Задача Fill in the missing number in the box

 20 + 15  2,015 =

A) 400 B) 4,000 C) 40,000

Problem Find the value of

A) 2,025 B) 2,020 C) 2,015

Problem There were 1,001 pieces of paper Some of them were cut into three parts Altogether, there are now 2,015 pieces of paper How many pieces were cut into three parts?

A) 507 B) 494 C) 1,014

Problem If the difference is 9,999 and the subtrahend is 1, the minuend is:

A) 9,998 B) 10,000 C) 1,000

Problem 10 A student only marked the odd numbered pages of his notebook by using only odd numbers such as 1, 3, etc He used 93 digits How many pages does the notebook have?

Problem 11 I chose a number and added it to Then I multiplied the resulting sum by After that I divided the resulting product by What is the number I chose, if the quotient is 4, and the remainder is 2?

Problem 12 The three following actions have been applied to the number in a random order: - multiplying by

- dividing by - adding

How many possible results are there?

Problem 13 There are less than 100 apples in a basket These apples can be divided equally between 2, 3, or children These apples can NOT be divided equally between children - one more apple would be needed to that What is the number of apples in the basket?

Problem 14 In ̅̅̅̅ ̅̅̅̅̅̅ each letter corresponds to a digit Identical letters correspond to identical digits and different letters correspond to different numbers What is the greatest possible number that corresponds to ̅̅̅̅̅̅̅̅̅̅?

Problem 15 How many missing addends are there in the expression ?

Problem 16 Which of the following is the smallest number: 62 345, 523 420 and 432 100?

Problem 17 How many pairs of integers which product is 63 can be selected from to 99?

Problem 18 What is the next number in the sequence of numbers? 2, 11, 20, 101, 110, 200, 1001, 1010, 1100, 2000, 10 001,

Problem 19 In the number A the positions of the tens and hundreds were exchanged and the resulting number was 1,234 What is the number A?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 В) 9,990 *,**0 ⇒9,990

2 С) 2015 < 2115, 2015 < 2215,…, 2015 < 2915

3 C) 778 1000 222 = 778

4 C) 81,000 The numbers are

5 С) 48 These are the sheets s of paper with page numbers (5; 6), …, (99; 100) Their total number is 48

6 C) 40,000

7 А) 2,025

8 А) 507 The number of pieces increases by a number that is the doubled number of the pieces that have been cut The number of pieces has increased by 1014 Therefore the number of pieces that have been cut is 507

Another way to find the answer is to check the given answers

9 B) 10,000 9,999+1=10,000

11 6 Let us solve the problem starting at the end: the number which we divided by is + = 14

The number that we multiplied by is 14 = The number that we added to is – =

12 3 If we mark the actions with M, D and A, there are ways for their sequential use: MDS, MSD, DMS, DSM, SMD, SDM

For each of these ways we obtain the following results: 4, 3, 4, 6, 4,

Thus the number of different results is 3: 3, and

13 90 The numbers divisible by and by with a remainder of are: 10, 20, 30, 40, 50, 60, 70, 80, 90 The ones divisible by are 30, 60 and 90 From the numbers 30+1, 60+1 and 90+1 only 91 is divisible by There are 90 apples in the basket

14 98,107 The sum ̅̅̅̅ is at most ̅̅̅̅ Therefore ̅̅̅̅̅̅̅̅̅̅

15 23 The addends are the one digit (except 0) and two digit numbers, divisible by The first number is 3, and the 33rd number is 99 The numbers that have been skipped are 33 – (7 + 3) = 23

16 62,345 62,345 < 432,100 < 523,420

17 3 The numbers are and 63; and 21; and

18 10,010 These are the numbers with as the sum of their digits They have been arranged from smallest to biggest The next number is 10,010

19 1,324 1,234 ⇒1,324

WINTER 2016: GROUP Problem What is the missing number?

A) 1,000 B) 1,010 C) 990

Problem Which three numbers are smaller than 30,020?

A) 30,019; 30,020; 30,021 B) 30,001; 30,010; 30,019 C) 30,001; 30,010; 31,000 Problem If the difference is 24,345, and the subtrahend is 6,707, the minuend is:

A) 31,052 B) 17,638 C) 17,648

Problem How many of the following expressions are correct? 165 + 561 = 727 264 – = 264 90,000 10 < 10,000

A) B) C)

Problem We have an apple, a pear, an orange and a lemon

We must distribute them among two children In how many different ways can we distribute the fruit, so that each child would get fruit?

A) B) C)

Problem What is the hundreds digit of the smallest 5-digit number, that has a sum of its digits equal to 25?

A) B) C)

Problem Which of the following numbers has as a digit of hundreds and as a digit of thousands?

A) 1,313 B) 3,311 C) 3,113

Problem Last year 33 white, red and yellow tullips blossomed at the same time in Maya’s garden The white and red tullips together were 19, and the red and yellow tullips together were 18 Which tullips were of the greatest number?

A) red B) yellow C) white

Problem The four-digit numbers smaller than 2,015 are:

A) 2,014 B) 1,015 C) 1,016

*1 +*96 92*

2016

A) B) C)

Problem 11 The natural numbers А, B, C and D are such that A × B = 2, B × C = and C × D = What is the number D?

Problem 12 Three friends weigh respectively 24, 30 and 42 kilograms They want to cross a river by using a boat that can carry a maximum of 70 kg At least how many times would this boat need to cross the river, so that all three of them would get to the opposite shore?

Problem 13 The numbers 1,001; 1,008; 1,015; 1,022; …; 2,016 are recorded according the following rule: we get each following number by adding to the preceding number, until we reach the number 2,016 How many numbers are there in total?

(Hint: 1,008 = 1,001 + 1 7; 1,015 = 1,001 + 7; 1,022 = 1,001 + 7, … ) Problem 14 In how many rectangles we find just one ant?

Problem 15 Place the numbers 1, 2, and in the squares in a way that would result in the greatest possible product What is the product?

Problem 16 I have baskets, each of which has 55 apples inside If I move the apples to 11 baskets, and each basket has the same number of apples, how many apples would there be in each basket? Problem 17 What is the digit of ones of the product of all odd one-digit numbers?

Problem 18 When A is divided by the remainder is What is the remainder when 3A is divided by 5?

Problem 19 What is the missing number?  = 330 + 330

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 А ? + 110 = 1,110, i.e ? = 1,000

2 В 30,001; 30,010; 30,019

3 А ? – 6,707 = 24,345 24,345 + 6,707 = 31,052

4 А 165 + 561 = 726, i.e the first expression is wrong

264 – = 1318; 264 = 792, i.e the second expression is wrong 90,000 10 = 9,000 < 10 000, i.e the third expression is correct

Of the three expressions, only the third one is correct

5 B There are possibilities If we number the fruit as 1, 2, and 4, then they would be distributed as follows:

First child Second child

1, 3,

1, 2,

1, 2,

2, 1,

2, 1,

3, 1,

6 А 25 = + + + + 1, therefore the smallest five-digit number with a sum of its digits equal to 25, is 10699

7 А In the number 1,313 the digit of ones is 3, the digit of tens is 1, the digit of hundreds is 3, and the digit of thousands is

plus the number of red tullips Therefore, the number of red tullips is

19 + 18 – 33 = 4, the number of white tullips is 15, and the number of yellow tullips is 14 The white tullips are of the greatest number

9 В The numbers from to 2,014 are 2,014 Among those numbers, the numbers from to 999 are not four-digit numbers Therefore the number we are looking for is 2,014 – 999 = 1,015

10 А We can solve the problem by carrying a check by using the possible answers We can start as follows:

1 + + * = …6, if * =

11 If A × B = ⇒ A = or A = If A = ⇒ B = ⇒ C = ⇒ D = If A = ⇒ B = ⇒ C = ⇒ D could not be a natural number such that × D =

Hence D =

12 Let C denotes the heaviest of the three friends, A - the lightest one, and B - the third one

It would be impossible for all three of them to cross the river in one go, because 24 + 30 + 42 = 96 > 70

Therefore the boat would have to return at least once, and the smallest possible number of river crossings would be

Following is an example of a way in which all three friends can cross the river to the opposite shore:

C stays on one of the shores, while A and B cross over to the opposite shore

A crosses back to the initial shore

A and C now cross to the opposite shore together

13 146 2,016 = 1,001 +  Therefore  = 1,015, i.e  = 145

АA1В, ВA1, А2, ВA2, ВA2С, A2C each

15 252 All possible products are:

1 43 = 86; 34 = 68; 24 = 72; 42 = 126; 23 = 92; 32 = 128; 14 = 84; 41 = 246; 13 = 104; 31 = 248;

3 12 = 144; 21 = 252 - the greatest possible product

16 25 The number of apples is 55 = 275 From 275 11 = 25, it follows that there are 25 apples in each basket

17 The number is one of the multipliers, and none of the multipliers is an even number Therefore the product must end in

18 First way: The numbers that when divided by leave a remainder of 2, are: 2, 7, 12, 17, 22,

The tripled numbers are: 6, 21, 36, 51, 66, When divided by 5, they all leave a remainder of

Second way: Since Dividend = Divisor Quotient + Remainder, we could present the numbers that when divided by leave a remainder of as follows: Quotient +

The tripled numbers could be presented as follows: 15 Quotient +

Therefore the remainder we are looking for would be equal to the remainder of dividing by 5, i.e the remainder =

19 198  = 330 + 330, therefore  = 396, i.e  = 198

SPRING 2016: GROUP Problem If  then  =:

A) B) C)

Problem –

A) 2,016 B) 4,032 C) other

Problem A hippo eats 200 kg of grass every day, which is three times less than the kilograms of grass that an elephant eats in a week How many days would it take for a hippo and an elephant to eat 2 tons of grass?

A) B) C)

Problem What is the natural number that is greater than the number which is greater by than 99,979 and smaller than the number which is times greater than 11,110?

A) 99,988 B) 99,989 C) 99,990

Problem The product of natural numbers is 72 The sum of these numbers is 15 and neither of them is In this case, which is the greatest among these numbers?

A) B) C)

Problem We have identical chocolate bars, each consisting of 28 pieces We have to divide them equally between children What is the minimum number of times we need to break each chocolate bar in order to this?

A) B) C)

Problem How many different three-digit numbers ̅̅̅̅̅ can you get as a product of the number 29 and a two-digit number?

A) B) C)

Задача A book has been numbered as follows: the pages of the first sheet have been numbered as and 2; the pages of the second - as and 4, and so on, until the pages of the last sheet, which have been numbered as 227 and 228 If I open the book at a random place, what is a possible product of the numbers of the two pages that I have opened the book at?

A) 9,900 B) 10,100 C) 90

Problem In the same room we have grandmothers, mothers, daughters and granddaughters What's the smallest number of people that there could be in the room?

A) B) C)

1, 12, 123,1234, , 12345678 and 123456789

A) B) C)

Problem 11 Four children met together: Adam, Bobby, Charley and Daniel Adam shook hands with of these children, Bobby shook hands with 2, and Charley shook hands with How many of the children’s hands did David shake?

Problem 12 A three-digit number consists of the digits 1, and The digit is not the digit of hundreds, and the digit is not next to the digit What is the number?

Problem 13 The product of a few different one-digit numbers is a number that is divisible by 10 (the remainder is 0), but is not divisible by 20 (the remainder is not 0) Which even numbers could be among the multipliers?

Problem 14 I placed addition sign and multiplication sign between the digits of the number 2016 Example: or How many different numbers will I get after calculating the expressions?

Problem 15 Annie has a magical necklace Each bead of the necklace is numbered (1, 2, 3, and so on) If between the beads numbered as and 15 there is the same number of beads, what is the total number of beads on Annie’s necklace?

Problem 16 In the garden of Rose there are 1,232 roses not yet in bloom and 1,168 that are blooming Every day new roses bloom and the ones that are already blooming not fade How many days will it take for the blossoming and non-blossoming roses to be an equal number?

Problem 17 A container, when full of water, weighs 994 kg and when half full, it weighs as much as empty containers How many kilograms does this container weigh when it is empty?

Problem 18 The natural numbers from 10 to 30 have been written on different cards (one on each card) What is the smallest number of cards that we would need to take without looking in order to make sure that there would be at least numbers divisible by 3?

Problem 19 What is the greatest possible number of different three-digit numbers that we can add and get a three-digit number as a result?

Problem 20 The expression we are going to use for this problem is

Problem Answer Solution

1 A  ⇒  ⇒  = 2 В –

3 А In one week a hippo can eat 1400 kg of grass, and an elephant can eat 600 kg of grass The amount they can eat in total is 2000 kg = tons

4 B The number which is greater than 99,979 by is 99,988, and the number which is times greater than 11,110 is 99,990 The number we are looking for is 99,989

5 В 15 = + + + = + + + = + + + Therefore the number we are looking for is

6 А The number of all pieces of all five chocolates is 28=140 Therefore each child should get 140 = 20 pieces

By breaking one chocolate, we can get 20 pieces and have left

In this way we can give 20 pieces to children, however there would be more children and more parts, each consisting of pieces, left

We can give parts with pieces each to each of the two children, and the fifth part, which consists of pieces, we can divide in parts of pieces The number of times we would need to break the chocolates is + = 7 B 29 17=493 <5**, 29 18=522, 29 19=551, 29 20=580,

29 21=609>5**, therefore there are options

8 B If I open the book, there will be two pages, both numbered The smaller number will be even, and the greater will be odd The numbers will be consecutive

9,900 = 99 100, 10,100 = 101 100, 90 = 10, therefore I have opened the book at the pages numbered as 100 and 101 A possible product is 10,100

9 B In order for one of the women to be a grandmother, she would need to have a daughter, and a granddaughter Therefore if there are two

grandmothers, who are also mothers, they have one daughter each, i.e daughters, each of whom is also a mother to granddaughter –

There are now daughters left, who are also mothers There are now mothers left, who are also grandmothers

10 C The digit we are looking for is the last digit of the sum of the last digits, i.e of + + + + + = 45

11 2 A B C D

A + + +

B + – +

C + – –

D + + –

If we add the number of hand shakes, the number must be divisible by 2, because each hand shake is counted twice

In this case the number of hand shakes is + x

We can mark the number of David’s handshakes with x The number x can NOT be greater than

6 + x can be divided by only if x is either or

However, x is not 0, because Adam shook hands with all the children Therefore x = David shook hands with children

12 312 or 213

The number is either *1* or **1

If the number is *1*, then the number is either 312 or 213

If the number is **1, then the digit and the digit are next to each other, which would mean that is not the digit of ones

13 2 or The product of the one-digit odd numbers is a number which ends in If we multiply it by or by we will get a number that ends in 0, but is not divisible by 20

14 4 016 ⇒ 0+16=16; 01+6=8 20 16 ⇒2+0 16=2; 20 1+6=26

201 ⇒2 + 01 = 8; 20 + = 26;

15 20 The beads with numbers from to 14 are situated between the beads with numbers from to 15 The beads are in total The beads on the opposite side are also If we also note the beads numbered and 15 we will find that the beads on Annie’s necklace are 9+2=20

by 32 roses This will happen in 32 3=8 days

17 142 The water in a half full container weighs as much as empty containers The water in a full container weighs as much as empty containers The water in a full container (the weight of the water + the actual container) weighs as much as empty containers

Therefore an empty container would weigh 994 = 142 kg

18 16 The numbers are in total of them are divisible by with a remainder of 0, are divisible by with a remainder of 1, and are divisible by with a remainder of

We would need to take + + = 16 cards, in order to make sure that we have taken cards that have such numbers on them which when divided by leave a remainder of

19 9 101 + 103 + 105 + 107 + 109 + 111 + 113 + 115 + 117 = 981; 101 + 103 + 105 + 107 + 109 + 111 + 113 + 115 + 117 +119 = 1100 20 2 In order for the value of the expression to be increased by 1, the following

needs to be true:

The first number in the expression + 12 – 10 needs to be exchanged with Then the initial value would be increased by

 2= would be possible if we exchange for = is not possible

The second number in the expression + 12 – 10 needs to be exchanged with 13 Then the initial value of the expression would be increased by  3= 13 is not possible

4 =13 is also not possible

The third number in the expression 3+12-10 needs to be exchanged with Then the initial value would be increased by

1  = 9, if we exchange 10 for  10=9 is not possible

We can exchange two of the numbers in the expression, so that the initial value would be increased by 1:

FINAL 2016: GROUP

Problem If ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅  , then  = ( )

A) B) C)

Problem A is a four-digit number Its digit of tens is two times greater than its digit of ones, its digit of hundreds is two times greater than its digit of tens How many A numbers with this property are there?

A) B) C) 18

Problem There are several points along a straight line A student placed a point between every two adjacent points After doing this times, there were now 33 points along the straight line How many points were originally on the straight line (before the student placed any extra points)?

A) B) C)

Problem An equilateral triangle with a side length of cm has been divided into smaller equilateral triangles with a side length of cm The numbers A, B, C, 4, 5, 6, 7, and have been placed inside the smaller triangles Now there are three equilateral triangles with sides of cm on the diagram and the sums of the numbers inside them are equal

What, then, is the greatest of those three numbers: A, B or C?

A) A B) B C) C

Problem Iva arrived at the bus stop and looked at her watch, which showed 08:01h It meant that she was minutes late for her bus What she did not know was that her watch was running minutes ahead If the bus came minute late, for how many minutes would Iva have to wait at the bus stop?

A) B) C) more than

Problem The number of astronomical hours in 2016 is equal to ( )

Problem A book is numbered as follows: the pages on the first sheet are numbered as and 2, the pages on the second sheet are numbered as and 4, and so on, until the last sheet, where the pages have been numbered as 127 and 128 If I tear off 11 consecutive sheets and add up all the page numbers of their 22 pages, which of the following sums is possible?

A) 255 B) 275 C) 341

Problem A storage room can be filled up with either 12 chests, or with 18 boxes There are currently chests and boxes in the room How many more chests can fit into the room?

A) B) C)

Problem A spring with a flow rate of 84 liters of water per minute provides water for three fountains Four times more water reaches the second fountain than does the first, and half as much water reaches the third fountain than does the second How many liters per minute is the flow rate of the fountain which receives the greatest amount of water?

A) 56 B) 48 C) 52

Problem 10 What is the three-digit number ̅̅̅̅̅, such that ̅̅̅̅̅ < 1116 and can be presented as the product of both 4, and of consecutive natural numbers?

A) 124 B) 120 C) 100

Problem 11 Find ̅̅̅ if

2 + 24 + 246 + 2468 + 24680 + 246808 + 2468086 + 24680864 + 246808642 = ̅̅̅̅̅̅

Problem 12 We can move from square A to square B by moving either horizontally or vertically from one square to another How many different routes that go through exactly squares are there?

B

A

Problem 14 Five people (A, B, C, D and E) are waiting in a queue C is between E and D, A is next to E, and B is NOT the last Which one is the last?

Problem 15 How many three-digit numbers ̅̅̅̅̅ are there, if 17 can divide both ̅̅̅ and ̅̅̅ without a remainder?

Problem 16 Seven numbers are arranged in a specific order and two of them are covered by shells How many times is the number under the first shell smaller than the number under the second shell?

1, , 2, 6, 24, , 720

Problem 17 At least how many of the numbers we need to change in order that the products of the numbers in the diagonals, rows and columns will be the same?

1

16

2

Problem 18 For a football game, the winner earns points and the loser earns points If the match is drawn, each team gets point After games, a team has earned 11 points What is the possible number of games that the team has lost?

Problem 19 A number is perfect when the sum of its divisors (except the number itself) equals the given number For example, the number is called perfect because it is equal to the sum of + + 3, where 1, and are all its divisors except the number itself The next perfect number is an even number greater than 24 and smaller than 30 What is the number?

Problem Answer Solution

1 C ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅  , so   would be a number digit of ones is Therefore the possibilities are either or

If we check both possibilities, we will find the correct answer: 2 C The numbers are one of two types: either ̅̅̅̅̅̅̅ or ̅̅̅̅̅̅̅

There are possibilities for the digit of thousands for each of the two numbers

Therefore there are 18 numbers with such property

3 A If the number of points is 2, then we will place There are now points, we will place another There are now points, we place another There are now 9, we place There are now 17 points, we place 16 There are now 33 points

4 B Let us compare the sums:

⇒ ⇒ ⇒ ⇒ Answer: B

5 А The watch showed that the time was 08:01h She was minutes late, which means that she was supposed to arrive at 07:59h, according to her watch This meant that she came minutes early because the bus (according to her watch) was supposed to arrive at 8:04h The bus was running minute late Therefore it would arrive at 8:05h Iva would have to wait for minutes

6 B 2016 is a leap year, so 7 C If I tear off sheets 1-11:

⏟

If I tear off sheets 2-12:

⏟

If I tear off sheets 3-13:

⏟

the other half It will take chests to fill half of the space Therefore there is room for another chests

9 B We can carry a check using the possible answers If the flow rate of the second fountain is 48 liters per minute, then the flow rates of the first and third fountains respectively would be 12 liters per minute and 24 liters per minute In this case 48 + 12 + 24 = 84

10 B We are looking for a three-digit number smaller than 124 The number is 120

In fact 120 < 124 and 120 = =

11 20 246808642

24680864 2468086 246808 24680 2468 246 24

Add all the digits of ones of these numbers: (2 + + + 8) = 40 is the digit of ones we are looking for and need to be added to the sum of all the digit of tens: (4 + + 8) + + = 42, is the digit of tens we are looking for

Therefore ̅̅̅ is equal to 20

12 10 , ,

c d B

b x y

a

A

13 211 The smallest sum is + + + 10 = 34, and the greatest sum is 50 + 20 + 10 + = 245

14 А or D First we can arrange the people as follows: DCEA or AECD Then B may be situated as follows:

ВDСЕA DСЕAВ АECDВ ВАECD

Since B is not the last, A or D can be the last

15 The number ̅̅̅ can be 17, 34, 51, 68 or 85 The number ̅̅̅ can be 17, 34, 51, 68 or 85 The second digit of ̅̅̅ is the first digit of ̅̅̅ Therefore the possibilities are the following: or We get the numbers 517, 685 and 851

16 120 Find the pattern first: after the first number 1, the rest are equal to the number before multiplying 1, 2, 3, 4, 5, respectively, i.e.:

1; 1=1; = 2; = 6; = 24; 24 5=120; 120 = 720 Therefore the numbers under the shells are and 120

120 120

17 If we change the number in the first row, first column, to 2, we will get a magic square

18 or 11 points can be earned in the following ways: wins, draws and defeats

or

3 wins, draws and defeats 19 28 The number is 28

28 = + + + + 14

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem The maximum number of people that can attend a party is @, among whom there can not be two people born in the same month Find @

Problem The sum of the digits, which are not part of the equation ̅̅̅̅̅̅̅̅̅ is # Find #

Problem Use & to represent the greatest possible product of two integers with a sum of # Find &

Problem There are 18 children in a class Each of them has either or balloons The total number of balloons is & The number of children who have balloons is § Find §

Problem There are § ways to travel from point X to point A There are * ways to travel from point

Problem Answer Solution

1 @ = 12 If 13 people are present, there would definitely be among them, who have been born in the same month (Dirichlet's Principle)

2 # =16 123456 12=1088 , therefore the missing digits are and

3 & = 64

16 = + 16 = + 15 = + 14 = + 13 = + 12 = + 11 = + 10 = + = + 8, therefore the possible products are 0, 15, 28, 39, 48, 55, 60, 63 and 64 The greatest is 64

4 § =

If all children are carrying balloons each, then 18 = 54 Therefore there would be 64 54=10 balloons left We would give those away to children In this way 13 children would be carrying balloons each, and children would be carrying balloons each

5 * = 60

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP

Problem Which one is the smallest product among the following?

A) B) C) D) Problem The number of integers from 98 to 1,000 which are divisible by is:

A) 301 B) 302 C) 303 D) 304

Problem If we have a 200 cm long tape and we cut off 12 dm from it, how long is the larger part of the two cut-offs?

A) 188 сm B) 80 сm C) dm D) 12 dm

Problem Peter reads 15 pages in 45 minutes How long will it take him to read 45 pages at that rate?

A) 15 mim B) h C) h 15 D) h 15

Problem The sum of the first 100 positive integers is 5,050 Find the sum of the first 100 positive odd integers

A) 10,000 B) 10,050 C) 10,100 D) 10,150

Problem There were a total of 90 coins in two boxes Ten coins were then shifted from the first box to the second As a result, the number of coins in the second box was twice as much as the number of coins in the first one What was the number of coins in the first box before the shift?

A) 100 B) 80 C) 60 D) 40

Problem In ̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ each letter corresponds to a digit Identical letters correspond to identical digits and different letters correspond to different numbers What is the greatest possible number that corresponds to ̅̅̅̅̅̅̅̅?

A) 1007 B) 1006 C) 1005 D) 1004

Problem What is the ones digit of the smallest natural number with sum of its digits equal to 2015?

A) B) C) D)

A) days B) 10 days C) 15 days D) 20 days Problem 10 How many squares are there in the figure below?

A) 12 B) 20 C) 22 D) 24

Problem 11 Find the value of

Problem 12 We have written down the numbers that are divisible by 5: 5, 10, 15, 20, 25, Underneath each of these numbers (in a second row) we have written down the sum of its digits Which place in the second row will be occupied by the number 14 for the first time?

Problem 13 When multiplying two numbers, Amy miswrote one of the factors: instead of 24 she wrote 42, and got a product of 714 What should be the correct product?

Problem 14 In a group of 60 people, 35 have brown hair, 30 have brown eyes and 20 have both brown hair and brown eys How many have neither brown hair nor brown eyes?

Problem 15 How many of the products of the numerical sequence

are divisible by 6?

Problem 16 A square is divided into squares The square R is coloured in red Each of the remaining squares is coloured either in red (R), blue (B) or green (G) If in each row and in each column the squares are coloured in all three colors, what would the colour of the X square be?

R

Х

G

Problem 17 In a certain year, January had exactly four Tuesdays and four Saturdays On what day did January fall that year?

Problem 18 111 111 111 divided by equals ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ Find the

Problem 19 If find

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C) 5 12 36 We compare

2 А) 301 The first number is , and the last is The number is

3 D) 12 200 – 120 = см; см < 12 dm

4 C) 45 can be read in 135 minutes, which is equivalent to hours and 15 minutes

5 A) 10,000

⏟

+ + +… + + = ⏟

Another way:

1+(2 +99)+3+(4+99 +…+ + +99) =5050 + 50 99=5050+4950=10 000

6 D) 40 After moving the coins, in the first box there are 30 coins, and in the second box there are 60 coins Before that there were 40 coins in the first box and 50 in the second box

7 D) 1004 If A<9, then ̅̅̅̅̅̅ ̅̅̅̅̅

In this case А= If B<8, then ̅̅̅̅̅̅ ̅̅̅̅̅<1000 In this case B=9 ̅̅̅̅̅̅ ̅̅̅̅̅ The possible values of C are the digits 0, 1, 2,3, 4, 5, and

+С would be a four digit number if C=3, 4, 5, and Only for C=5, and 7, we get the four digit number ̅̅̅̅̅̅̅̅ The greatest of them is , which we get from С=

8 A) The smallest number consists of the smallest amount of digits, therefore the predominant digits are

From 2015 9= 223 (remainder 8), it follows that the smallest number is ⏟

9 С) 15 The minutes in hours are The hours in a day are 24

360 24 = 15

10 B) 20 12 squares with a side of 1; squares with a side of 2; squares with a side of 3;

11

396

⏟

=

=

=12 12 19 5, ⏟

13 408 714 42 = 17; 17 24 = 408

14 15 With brown hair, but not with brown eyes: 35 – 20 = 15; With brown eyes, but not with brown hair: 30 – 20 = 10 With brown eyes and with brown hair: 20

We need to deduct 15 + 10 + 20 = 45 from the total number, in order to find the number of people that have neither brown hair nor brown eyes: 60 – 45 = 15

15 98 All products are divisible by It is known that the product of two consecutive numbers is divisible by 2; of three consecutive numbers – by , of four consecutive numbers –

by etc 16 Red

or green

R B G R G B

B G R B R G

G R B G B R

Wednesday, Thursday and Friday January is on Wednesday

Other way

If January is: On Monday:

The Mondays would be – 5; Tuesdays – 5, Wednesdays – 5; Thursdays – 4, Fridays – 4; Saturdays – 4, Sundays – 4; On Tuesday:

The Mondays would be – 4; Tuesdays – 5, Wednesdays – 5; Thursdays – 5, Fridays – 4; Saturdays – 4, Sundays – 4; On Wednesday:

The Mondays would be – 4; Tuesdays– 4, Wednesdays – 5; Thursdays– 5, Fridays– 5; Saturdays – 4, Sundays – 4; On Thursday:

The Mondays would be – 4; Tuesdays – 4, Wednesdays – 4; Thursdays – 5, Fridays– 5; Saturdays – 5, Sundays – 4; On Friday:

The Mondays would be – 4; Tuesdays – 4, Wednesdays – 4; Thursdays – 4, Fridays – 5; Saturdays – 5, Sundays – 5; On Saturday:

The Mondays would be – 5; Tuesdays – 4, Wednesdays – 4; Thursdays – 4; Fridays – 4; Saturdays– 5, Sundays– 5; On Sunday:

The Mondays would be – 5; Tuesdays – 5, Wednesdays – 4; Thursdays – 4; Fridays – 4; Saturdays – 4, Sundays – 5;

18 9 12345679 9=111 111 111

19 1

WINTER 2016: GROUP Problem

A) B) C) D) other

Problem Which of the following products is the greatest?

A) B) C) D)

Problem The sum of five different odd natural numbers is 27 Which of these numbers is the greatest?

A) B) C) D) 11

Problem How many even numbers are there from 205 to 2,017?

A) 1,812 B) 1,813 C) 907 D) 906

Problem If we diminish the dividend 10 times and increase the divisor 10 times, what would happen to the quotient?

A) it will be diminished 100 times B) it will not change C) it will be diminished 10 times D) it will increase 10 times

Problem By how many are the non-coloured squares less than the coloured squares?

A) B) C) D)

Problem Find the difference of the smallest number that is greater than 2,016 and has the same sum of its digits as 2,016, and the number 2,016

A) B) C) D)

Problem Steve had a bowl with some sweets in it At first he ate a third of the sweets After that he ate a fourth of what was left in the bowl In the end, he ate a sixth of the remaining sweets At this point there were 10 sweets left in the bowl How many sweets were there in the beginning?

A) 27 B) 24 C) 21 D) 18

Problem What is the sum of the missing digits in the following equation?

A) 29 B) 27 C) 24 D) 18

A) B) C) 10 D) 15

Problem 11 The natural number A would be increased 11 times if, on its right side, we write down one of the following nine digits: 1, 2, 3, 4, 5, 6, 7, or How many digits does the number A have?

Problem 12 In how many different ways can we divide a set of different weights (from to grams each) in two groups of equal weight?

Problem 13 Place the digits 0, 1, and in the squares in such a way that would result in the greatest possible product What is the product?

 

Problem 14 When x is divided by 55 the remainder is 22 What is the remainder when is divided by 55?

Problem 15 A rectangle has a width of 18 cm, and a length four times greater than the width How many dm is the parameter of the rectangle?

Problem 16 – = ?

Problem 17 One of the three brothers A, B and C took the golden apple Their father asked them who took it and they answered as follows:

A: “B took the golden apple.” B: “I took the golden apple.” C: “A took the golden apple.”

Who actually took the golden apple, if only one of the three brothers was telling the truth?

Problem 18 How many odd natural numbers that are smaller than 15 can be presented as a sum of two prime numbers?

(Hint: A prime number is a number, larger than 1, that can only be divided evenly by itself and For example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, )

Problem 19 There were 18 apples in one basket, and 20 apples in another basket I took a few apples from the first basket, and then I took as many apples as were left in the first basket, from the second basket How many apples in total are left in both baskets?

Problem 20 With a single jump, a grasshopper can move by cm and mm, and with

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B First way: 892 440 448=892 888 =

Second way: (223 110 112) = = 2 C 123 > 123

123 < 123

123 = 123 42 > 123 40 = 123 The greatest product is

3 D + + + + = 25, therefore the numbers are 1, 3, 5, and 11 The greatest of these numbers is 11

4 D First way:

Let us observe the pairs of numbers – odd, even (205, 206); (207, 208);…; (2015, 2016); (2017, 2018)

The pairs are 907 In each of them there is an even and an odd number The odd numbers are 907, and the even numbers from 205 to 2017 are 906

Second way:

The even numbers from 205 to 2017 can be presented as follows: 206 + 2; 206 + 2; 206 + 2; ; 206 + 905 = 2016 5 A (a 10) (b 10) = (a b) 100

6 D The coloured squares are 16

The non-coloured squares are 14: squares with a size of 1, squares with a size of 2, square with a size of 3

7 A The smallest number we are looking for is 2,025 The difference we are looking for is

8 B We can reach the solution by carrying out a check by using the possible answers

If the sweets are 24, then Steve would have eaten sweets, and there would have been 16 sweets left

After that he would have eaten 4, and 12 sweets would have been left In the end he would have eaten 2, and 10 sweets would have been left

10 B The digit b is equal to The digit a is equal to 3, 4, 7, or

11 1 If we present the number as ̅̅̅̅ x being one of the digits from to 9, then 10A + х = 11A Therefore x=A, i.e A is a one-digit number

12 4 The total weight of the set of weights is 28 grams If the weight of grams is in the first group, then we would need to add other weights in the group that have a total weight of grams = + = + = + = + + 4, therefore we could group them in ways

First way: first group – 7, 6, 1; second group – 2, 3, 4, Second way: first group – 7, 5, 2; second group – 1, 3, 4, Third way: first group – 7, 4, 3; second group – 1, 2, 5, Fourth way: first group – 7, 4, 2, 1; second group – 3, 5, 13 630 30 21 = 630

14 11 The number is 55A + 22

Therefore is 165A + 66 = 55 (3A+1) + 11

15 18 The length is 72 cm The parameter is 180 cm = 18 dm 16 252 (999 + 111 102) = 252

17 А A and B claim the same thing From the condition of the problem it follows that they are not telling the truth Only C is telling the truth

18 4 = + 3; = + 2; = + 2; 13 = 11 +

19 20 First way: If we take apples, then the apples left in the first basket would be 16 Then if we take 16 apples from the second basket, apples will be left there The total number of apples left is 20

Second way: If we take x number of apples from the first basket, then the apples left in it would be 18 x Then if we take 18 x from the second basket, there would be 20 (18 x) = + x apples left in it

The total number of apples left is 18 – x + + x = 20.

20 81 The grasshopper will go a distance of 81 mm, and the turtle would go a distance of x mm

SPRING 2016: GROUP

Problem The tens and hundredths digits in the number A were reversed and the resulting number turned out to be 20.16 What was the original number?

A) 60.12 B) 61.02 C) 20.61 D) 10.26

Problem Instead of being reduced 10 times, a number was increased 10 times and the result was 20.16 What is the number that was meant to be received?

A) 201.6 B) 2.016 C) 0.2016 D) other

Problem A is the smallest natural number, so that when divided by leaves a remainder of What would be the remainder if the number A is divided by 4?

A) B) C) D)

Problem The speed of a boat going along the stream is 18 km/h, and the speed of the same boat going against the stream is 12 km/h What is the speed of the boat in still water?

A) 13 km/h B) 14 km/h C) 15 km/h D) 30 km/h

Problem Calculate the value of the expression

A) 0.375 B) 0.275 C) 0.125 D) 0.1

Problem The number of positive integers from to 2,016, which cannot be divided by or 5, is:

A) 1210 B) 1008 C) 202 D) 806

Problem How many proper irreducible fractions are there, which have a one-digit denominator and a numerator other than 0?

A) 25 B) 27 C) 30 D) 35

Problem If the square is magical, find

X

Problem The number ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ consists of 12 digits (1, 2, 3, , and the number is included times), and it is divisible by and What is the digit ?

A) B) C) D)

Problem 10 The rectangle ABCD below is made up of five identical rectangles How many square centimeters is the area of the rectangle ABCD equal to, if BC = 1.5 cm?

A) 3.75 B) 4.75 C) 3.5 D)

Problem 11 If + 12 + 123 + 1,234 + + 12,345,678 + 123,456,789 = ̅̅̅̅̅̅̅̅̅ , then ̅̅̅̅̅ Problem 12 We have identical chocolate bars, each consisting of 28 pieces We have to divide them equally between children What is the minimum number of times we need to break each chocolate bar in order to this?

Problem 13 The natural number A has divisors (natural numbers, including and the number

A itself), the natural number B has divisors (natural numbers, including and the number B itself), and the smallest common denominator of the two numbers is How many natural numbers are there which are divisors of the number equal to A + B (including and the number itself)?

Problem 16 Four children met together: Adam, Bobby, Charley and Daniel Adam shook hands with of these children, Bobby shook hands with 2, and Charley shook hands with How many of the children’s hands did David shake?

Problem 17 In a sports club there are 12 gold, 14 silver and 13 bronze medalists There are 30 individual medalists in total in the club and each of them has at least one medal None of the gold medalists has a silver medal, but of them have bronze medals too How many of the bronze medalists also have silver medals?

Problem 18 Annie has a magical necklace Each bead of the necklace is numbered (1, 2, 3, and so on) If between the beads numbered as and 15 there is the same number of beads, what is the total number of beads on Annie’s necklace?

Problem 19 The expression we are going to use for this problem is

Replace one of the numbers from the expression with such a number that the initial value of the expression would be increased by How many of the numbers can be changed?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 A In the number 20.16 the digit of tens is 2, and the digit of hundredths is Therefore the number we are looking for is 60.12

2 C The number 20.16 is 10 times greater than the given number Therefore the given number is 20.16 10 = 2.016 If we decrease the number 10 times we would get 2.016 10 = 0.2016

3 C The smallest natural number which, when divided by leaves a remainder of 6, is Therefore = (remainder of 2)

4 C The sum of 18 and 12 is equal to the doubled speed of the boat in still water

Therefore the speed of the boat in still water is equal to 30 = 15 km/h

5 A

6 В The natural numbers from to 2016 are 2016

1008 of them are even, and 202 end in the - numbers 1.5, 3.5, 5.5, 7.5, 401.5, 403.5

The numbers which are divisible either by 2, or by 5, are 1210 in total The numbers which are NOT divisible neither by 2, nor by 5, are 806

7 В The number of all proper fractions with one-digit denominators is + + + + + + + = 36

The reducible fractions among them are: The number of irreducible fractions is 36 – = 27 in total

Y X

⇒

2

3

1

9 A The number would be divisible by if a = or = The number would be divisible by if a =

That is only possible if =

10 A The sides of each of the five identical rectangles are 1.5 cm and

1.5 ThenAB = 0.5 + 1.5 + 0.5 = 2.5 cm andthe area of the rectangle ABCD = 1.5

11 205 The last digit is 5, because the sum of 1, 2, 3, 4, and is 45

The digit before last would be the last digit of the sum 1+2+3+ +7+8 plus 4, i.e 36 + = 40 The digit before last is

The other digit we are looking for is equal to the last digit of 1+2+3+ +7 plus 4, i.e 28 + = 32 The last three digits are 2, and ̅̅̅̅̅ 205

12 6 The number of all pieces of all five chocolates is 28=140 Therefore each child should get 140 = 20 pieces

By breaking one chocolate, we can get 20 pieces and have left

In this way we can give 20 pieces to children, however there would be more children and more parts, each consisting of pieces, left

13 6 The number B is a simple number and is a divisor of Therefore B = and A = 9, A + B = 12, and the natural numbers which are divisors of 12 are 6: 1, 2, 3, 4, and 12

14 5 The numbers smaller than 2016 are 6: 1026, 1062, 1206, 1260, 1602, 1620;

The numbers greater than 2016 are 11: 2061, 2106, 2160, 2601, 2610, 6012, 6021,6102, 6120, 6201, 6210

15 3 The possible options are: and 5, product 20; and 7, product 42; and 9, product 72

16 2 A B C D

A + + +

B + – +

C + – –

D + + –

If we add the number of hand shakes, the number must be divisible by 2, because each hand shake is counted twice

In this case the number of hand shakes is + x

We can mark the number of David’s handshakes with x The number x can NOT be greater than

6 + x can be divided by only if x is either or

However, x is not 0, because Adam shook hands with all the children Therefore x = David shook hands with children

17 4 Let us subtract 12 gold medalists from the total number of medal winners Therefore 18 people have won bronze and silver medals

The bronze medalists who have no gold medals are 13 – = The silver medalists are 14 We get 14 + = 22 in total

18 20 The beads with numbers from to 14 are situated between the beads with numbers from to 15 The beads are in total The beads on the opposite side are also If we also note the beads numbered and 15 we will find that the beads on Annie’s necklace are + = 20

19 2 In order for the value of the expression to be increased by 1, the following needs to be true:

The first number in the expression + 12 – 10 needs to be exchanged with Then the initial value would be increased by

 = would be possible if we exchange for  = is not possible

The second number in the expression + 12 – 10 needs to be exchanged with 13 Then the initial value of the expression would be increased by  = 13 is not possible

4  = 13 is also not possible

The third number in the expression 3+12-10 needs to be exchanged with Then the initial value would be increased by

1  = 9, if we exchange 10 for  10 = is not possible

We can exchange two of the numbers in the expression, so that the initial value would be increased by 1:

8 + – 10 + –

20 2 We would have to place coins on each pan of the scales

If the scales are balanced, then we would have to place of the remaining coins on each pan of the scales and if the scales are still balanced, the third remaining coin is fake; if it is not balanced, the lighter (fake) coin can be found on the higher pan of the scales

FINAL 2016: GROUP

Problem Find the sum of the fractions and , if 

A) B) C) D) or

Problem The numbers from to 40 are written down one after another: 01234567891011 37383940

In how many ways can we pick out two consecutive digits, so that their sum would be 10?

A) B) C) D)

Problem There are several points along a straight line A student placed a point between every two adjacent points After doing this a number of times, there were now 129 points along the straight line How many points were possibly on the straight line originally (before the student placed any extra points)?

A) B) C) D)

Problem Adam has 44 marbles - blue, red, white and yellow The number of the blue marbles is more than that of the red, the number of the red marbles is more than that of the white, and the number of the white marbles is more than that of the yellow How many blue marbles does Adam have?

A) 12 B) 14 C) 16 D) 18

Problem Iva arrived at the bus stop and looked at her watch, which showed 08:01h It meant that she was minutes late for her bus What she did not know was that her watch was running minutes ahead If the bus came minute late, for how many minutes would Iva have to wait at the bus stop?

A) B) C) D)

Problem How many four-digit numbers are there that can be written down using the four digits 1, 2, and 4, in such a way that is not the digit of ones, is not the digit of tens, is not the digit of hundreds, and is not the digit of thousands?

Problem What is the number in which the digit of tenths is smaller than the digit of tens?

A) 222.31 B) 209.09 C) 32.32 D) 345.255

Problem A storage room can be filled up with either 12 chests, or with 18 boxes There are currently chests and boxes in the room How many more boxes can fit into the room?

A) B) C) D)

Problem A spring with a flow rate of 84 liters of water per minute provides water for three fountains Four times more water reaches the second fountain than does the first one, and half as much water reaches the third fountain than does the second one How many liters per minute is the flow rate of the fountain which receives the least amount of water?

A) B) C) 12 D) 14

Problem 10 Which of the following fractions is equal to ? A)

B)

C)

D)

Problem 11 I added each two of the numbers A, B and C, and then added the sums again At last I got What is A + B + C equal to?

Problem 12 If the dividend is , and the divisor is

, then the quotient is a number that

is written down using X different digits Calculate X

Problem 14 Ten teams are participating in a football tournament Each team plays every other team exactly once For each match, the winning team gets points, the losing team gets points, and in case of a draw each team gets point At some point in the tournament it turns out that the teams have earned a total of 131 points How many games are yet to be played?

Problem 15 The two-digit numbers ̅̅̅, ̅̅̅ and ̅̅̅ are multiples of 17 (each letter represents one number) Calculate the greatest possible value of

Problem 16 How many of the four-digit numbers written down using the four digits 2, 0, 1, are divisible by 36?

Problem 17 A cuboid has been formed from two identical cubes with a surface area of 1.5 sq

cm each Calculate the surface area of this cuboid

Problem 18 The fraction is presented as an infinite repeating decimal What are the digits that are not used when writing it down?

Problem 19 How many prime numbers are there, smaller than 99, for which is also a prime number smaller than 99?

Problem 20 The numbers A and B are such that 143 A + 325 B = 6.5 Calculate the value of the following expression if A and B are the same as above:

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 D  ⇒ or 

2 C

0123456789101112131415161718192021222324252627282930313233343 53637383940

The digit couples are: 91, 19, 28, 82, 37, 73

3 B

If the number of the points is 3, we will place another 2; now there are points and then we will place more points to get Then we will place another points to get 17 Then we will place another 16 to get 33 Then we will place another 32 to get 65 Then we will place another 64 and the final number of the points will be 129

4 C

If the yellow marbles are 4, then the white marbles are 10, the red are 14, and the blue are 16 The total number is + 10 + 14 + 16 = 44

5 А

The watch showed that the time was 08:01 h She was minutes late, which meant that she was supposed to arrive at 07:59 h, according to her watch This meant that she came minutes early because the bus (according to her watch) was supposed to arrive at 08:04 h The bus was running minute late Therefore it would arrive at 08:05 h Iva would have to wait for minutes

6 B

We can choose either 1, or for the digit of thousands 1, or for the digit of hundreds; 1, or for the digit of tens and 2, or for the digit of ones

If the digit of thousands is 3, then the numbers are 3412, 3214 and 3142 The number of four-digit numbers are in total

7 D

In the number 345.255, the digit of tenths is and is smaller than the digit of tens, which is

8 D

of the room is occupied The unoccupied part is equal to ,

which means there is still space for boxes

9 C

We can carry out a check using the possible answers If the flow rate of the second fountain is 48 liters per minute, then the flow rates of the first and third fountains would respectively be 12 liters per minute and 24 liters per minute In this case 48 + 12 + 24 = 84

10 A

The greatest common divisor of 4095 and 6426 is 63 After cancelling the fraction, we would get

as a result

11 or

1.2 ⇒

12 The quotient is 12 345 679

13 1/3

Alex is able to pay (1 + + + 10 + 20 + 50) = 264 cents

3.96 – 2.64 = 1.32, so his father needs to pay 132/396=1/3 of the book price

14 or

The total number of matches is 45 and the maximum points a team can earn is 135 Now since 131 points have been earned, there are points left Therefore 131 points can be earned in three ways:

15 21

The number ̅̅̅ can be 17, 34, 51, 68 or 85 The number ̅̅̅ can be 17, 34, 51, 68 or 85 We must keep in mind that the second digit of ̅̅̅ is also the first digit of ̅̅̅ Therefore the options are either or

1 ⇒ ⇒ ⇒ ⇒

The sum we are looking for is 21

16

The sum of the digits is 9, so all of the four-digit numbers written down using these digits are divisible by In order for it to be divisible by 36, it also needs to be divisible by In this case the last two digits must be arranged as follows: хх6 , хх2 , хх and хх It is now possible to find the four-digit numbers we are looking for: 1260, 2160, 6120, 1620, 2016 and 6012

17 2.5

The length of the edge of the cube is 0.5 cm We can find the surface area of the cuboid by subtracting the doubled area of one of the sides of the cube from the sum of the surface areas of the two cubes: 1.5 – 0.5 0.5 = – 0.5 = 2.5 sq cm

18 3, and

̅̅̅̅̅̅̅̅̅̅

The digits 3, and are not used to write down the repeating decimal

19 (2; 5), (3; 7), (5; 11), (11; 23), (23; 47), (29; 59), (41; 83)

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem Subtract the number @ from the numerator and from the denominator of the fraction

, in order to get a fraction equal to Calculate @

Problem We have made a random selection of # three-digit numbers Among all three-digit numbers, there would always be at least which are co-prime with @ Find the smallest possible value of #

Problem The children from a school class had to solve # problems for homework Three of them solved respectively 60, 50 and 40 problems At least & problems have been solved by all three of them Find &

Problem Two ants started walking towards each other simultaneously from the two points A and B One of the ants will travel the distance in & hours and the other one will travel the distance two hours faster What part of the full distance would they need to walk before they meet, if two hours have passed since their departure? Denote the answer using § Find §

Problem If §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ =

2 # = 72

The numbers which are not coprimes to are 69 Those numbers are: = 4, = 7, = , , 76 = 988

The number we are looking for in this case is 69 + = 72

3 & =

The first student did not solve problems, the second did not solve 22, and the third did not solve 32 problems If none of the other two students solved the unsolved problems, then the unsolved problems would be 66 in the worst case scenario In this case 72 – 66 = problems have been solved by the three of them

4 § = They have walked

of the full distance

The remaining distance they have yet to walk is

5 * =

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP

Problem 1,000,000 – 100.1 =

A) 999,998.9 B) 999,989.99 C) 900,000.9 D) 1,000,010.01 Problem Find the value of the expression

A) B) C) 0,9 D) other

Problem When dividing the sum of consecutive odd numbers by 6, the remainder we get is always:

A) B) C) D)

Problem If the three-digit numbers ̅̅̅̅̅̅ and ̅̅̅̅̅ is divisible by 9, what is the number Y?

А) B) C) D)

Problem Peter reads 15 pages in 20.5 minutes How long will it take him to read 16 pages at that rate?

A) 21 33 sec B) 21 52 sec C) 20.33 D) 21.52

Задача In the table the numbers are placed so that all rows, columns and the two diagonals will sum to the same value Which number you should place instead of „?“ ?

1.2

?

Problem Two fractions divide the interval with the boundaries of and into three equal parts The smaller of the two fractions is:

A) B) C) D)

Задача The price of a product has been changed twice, either increased or decreased In which case we can buy the product at the lowest price?

A) If first decreased by 10%, then increased by 10% B) If first increased by 15 %, then decreased by 15 % C) If first increased by 20 %, then decreased by 20 % D) If first decreased by 25 %, then increased by 25%

Problem Find the tens digit of the value of the expression

A) B) C) D)

Problem 10 In a group of 80 people, 39 have brown hair, 30 have brown eyes and 15 have both brown hair and brown eys How many have neither brown hair nor brown eyes?

A) 15 B) 26 C) 31 D) other

Задача 11 All four-digit numbers divisible by are formed by the digits 0, 2, and Determine the number of possible values of x + y + z, if x, y and z are positive integers

(Hint: ⏟ ⏟ ⏟ )

Problem 12 There were a total of 90 coins in two boxes A third of the coins from the first box were then shifted into the second one As a result, the number of coins in the second box was twice as much as the number of coins in the first one What was the number of coins in the first box before the shift?

Problem 14 Find the value of the expression

Problem 15 A rectangular sheet of size cm by cm is cut into squares with side lengths that are whole numbers If the sheet is cut into the smallest possible number of squares, how many squares of side length cm are there?

Problem 16 How many of the products of the numerical sequence

are divisible by 24?

Problem 17 How many are the 4-digit numbers which end in and are divisible by 3?

Problem 18 In a certain year, February had exactly five Saturdays On what day did March fall that year?

Задача 19 If then the value of is…

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B

2 B

3 D We can easily find the answer by observing an example:

1 + + = 9; = 1(remainder of 3)

If the first odd number is 2n+1, the next two would be 2n+3 and 2n+5 Their sum is 6n+9 The quotient is 2n+1, remainder –

4 D The sum of the digits of ̅̅̅̅̅ is a number that is a multiple of 9, if X = The sum of the digits of the number ̅̅̅̅̅̅ for Х = is + Y is a number that is a multiple of 9, if Y =

5 B 15 pages in 20,5 minutes = 1230 seconds One page can be read in 82 seconds = minute and 22 seconds

16 pages = 15 pages +1 page

16 pages can be read in 20 minutes and 30 seconds + minute and 22 seconds = 21 minutes 52 seconds

6 C Let x denote the number n the second row, third column We reach the conclusion that:

7 A From ; it follows that the interval is If we divide it into three parts, we will find that each part is

The smaller of the two fractions is

8 D The price of the goods is respectively 0,99; 0,9775; 0,96; 0,9375 of the initial price

9 A

The number is a multiple of 100, therefore the digit of tens is 10 B With brown hair, but not with brown eyes: 39 – 15 = 24;

With brown eyes, but not with brown hair: 30 – 15 = 15 With brown eyes and with brown hair: 15

number of people who neither have brown hair nor brown eyes: 80 – 54 = 26

11 2 The sum of the digits of each of the numbers we are looking at is 12 The number is divisible by 3, but not by Therefore y =

number x, y, z x+ y+ z

2370 790=3 194=2 97 x=1 ; y=1; z=1

3

2730 910 =3 182=2 91 x=1 ; y=1; z=1

3

3270 3270=3 218=2 109 x=1 ; y=1; z=1

3

3720 3720=2 124=8 31 x=3 ; y=1; z=1

5

7230 7230=2 241=2 241 x=1 ; y=1; z=1

3

7320 7320=2 244=8 61 x=3 ; y=1; z=1

5

12 45 It is a good idea to solve the problem by starting at the end

The coins at the end are respectively 30 and 60, in the first and second boxes 2/3 of the initial coins are left in the first box Therefore the coins in the first box were 45

13 5 Solution The position of the points is ABC, ACB, BCA or CBA The first position is not possible, because AB=16cm>AC=8cm CBA is not possible either because AB=16 cm>AC=8 cm If the position is ACB, then the distance between the midpoints of BC and AB is 5cm

If the position is BCA, then the distance between the midpoints of BC and AB is 5cm

14 36

⏟

squares with a side of 1cm

16 97 The product of consecutive numbers is divisible by 17 300 The digit numbers that end with would be divisible by if the digit

number that we got after removing the digit of ones of the digit number is divisible by The total number of digit numbers is 999 – 99 = 900 Each third number is divisible by

Their number is 300 18 Sunday If 1st of February is on a:

Monday:

The Mondays would be – or 5; Tuesdays – 4, Wednesdays – 4; Thursdays – 4, Fridays – 4; Saturdays – 4, Sundays – 4;

Tuesday:

The Mondays would be – 4; Tuesdays – or 5, Wednesdays – 4; Thursdays – 4, Fridays – 4; Saturdays– 4, Sundays – 4;

Wednesday:

The Mondays would be – 4; Tuesdays –, Wednesdays – or 5; Thursdays – 4, Fridays– 4; Saturdays – 4, Sundays– 4;

Thursday:

The Mondays would be – 4; Tuesdays – 4, Wednesdays – ; Thursdays – or 5, Fridays – 4; Saturdays – 4, Sundays – 4;

Friday:

The Mondays would be – 4; Tuesdays – 4, Wednesdays – 4; Thursdays – 4, Fridays – or ; Saturdays – 4, Sundays – 4;

Saturday:

The Mondays would be – 4; Tuesdays – 4, Wednesdays – 4; Thursdays– 4, Fridays– 4; Saturdays – or 5, Sundays – 4;

Sunday:

The Mondays would be – 4; Tuesdays – 4, Wednesdays – 4; Thursdays – 4, Fridays – 4; Saturdays – 4, Sundays – or 5;

19

( )

WINTER 2016: GROUP Problem Which of the following equalities is correct?

A) B) C) D) Problem What is the value of the expression ?

A) B) C) D)

Problem The value of the expression | | | | | | | | | | | | | | | | is equal to:

A) 36 B) C) D)

Problem How many even numbers are there with an absolute value smaller than 10?

A) B) C) D) other

Problem What is the remainder left when is divided by 15?

A) B) C) 10 D) 15

Problem The product of two integers smaller than and greater than ( 77) is 77 The sum of these numbers is:

A) B) C) 18 D) 78

Problem When apples are dried they lose 84% of their weight How many kilos of apples would be needed to produce 24 kg of dried apples?

A) 84 B) 100 C) 125 D) 150

Problem Steve had a bowl with some sweets in it At first he ate a third of the sweets After that he ate a fourth of what was left in the bowl In the end, he ate a sixth of the remaining sweets The initial number of sweets CANNOT be:

A) 12 B) 24 C) 30 D) 36

Problem The code for a safe is made up of all digits that are multiples of 3, without any digit being repeated.What is the maximum number of failed attempts that we would need to make before breaking the code?

A) B) C) 23 D) 24

Problem 10 Two identical rectangles with a length of cm and a width of cm overlap, forming a square How many sq cm is the area of their common part (the part where they overlap)?

Problem 11 The natural number A would be increased 11 times if we write down on the right side of it one of the following nine digits: 1, 2, 3, 4, 5, 6, 7, or How many digits does the number A have?

Problem 12 In how many different ways can we divide a set of different weights (from to grams each) in two groups of equal weight?

Problem 13 The natural numbers from to N have been recorded next to one another The result is a multi-digit number containing N digits What is the number N?

Problem 14 What is the smallest natural number N, for which the product of 13, 17 and N can be presented as the product of three consecutive natural numbers?

Problem 15 Calculate the following:

Problem 16 Three fishermen go fishing regularly The first one goes every day, the second one goes once every three days, and the third one goes once every four days Today is Sunday and they’re all at the lake, fishing In how many days, counting from Monday onwards, are they all going to be together at the lake again?

Problem 17 One of the three brothers A, B and C took the golden apple Their father asked them who took it and they answered as follows:

A: “B took the golden apple.” B: “I took the golden apple.” C: “A took the golden apple.”

Who actually took the golden apple, if only one of the three brothers was telling the truth?

Problem 18 What is the 2016th digit after the decimal point of the infinite periodic decimal equal to the fraction

?

Problem 19 There were 18 apples in one basket, and 20 apples in another basket I took a few apples from the first basket, and then I took as many apples as were left in the first basket, from the second basket How many apples in total are left in both baskets?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C + = ≠

(the only correct equality)

(division by is not possible)

2 B

3 D 1 + – + – +

4 C The numbers are 8, 6, 4, 2, 0, 2, 4, и

5 C quotient + remainder

Hence the remainder is a natural number that is a multiple of 5, but is smaller than 15, i.e it is either 0, or 10

The remainder is not 0, because is not divisible by

The remainder is not 5, because in this case The number equal to = ⏟

is not divisible by

Therefore the remainder is 10

6 А 77 = ( 1) ( 77) = ( 7) ( 11) =

Among these, two integers are smaller than and greater than 77, hence the numbers we are looking for are and 11 Their sum is 18

7 D 100 % 84 % = 16 % 16% of x is 24 kg

We need 150 kg of apples to produce 24 kg dried apples

by 12 (the least common multiple of 3, and 6) The only number that is not divisible by 12 is 30

9 C The multiples of are the digits 0, 3, and Therefore all possible codes would be = 24

In the worst case scenario, we would break the code after making 23 failed attempts The code would be the 24th possible number

10 B The square must have a side of cm Then the common part would be a rectangle with a width of cm and a length of cm

The area would then be sq cm

11 1 If we write down the number ̅̅̅̅ where x is one of the digits from to 9, then 10A + x = 11A Then x = A, i.e A is a one-digit number

12 4 The total weight of the set of weights is 28 grams If the weight of grams is in the first group, then we would need to add other weights in the group that have a total weight of grams = + = + = + = + + 4, therefore we could group them in ways

First way: first group – 7, 6, 1; second group – 2, 3, 4,

Second way: first group – 7, 5, 2; second group – 1, 3, 4,

Third way: first group – 7, 4, 3; second group – 1, 2, 5,

Fourth way: first group – 7, 4, 2, 1; second group – 3, 5,

13 1,107 + 90 + 900 + (N 999) = 3N

N = 1,107

14 600 13 17 = 52 51, therefore the smallest natural number we are looking for is 50 = 600

13 17 600 = 50 51

16 11 Let us assume that today is Sunday and all three fishermen are fishing together The first fisherman goes fishing every day

The second fisherman will go fishing on Wednesday, Saturday, Tuesday, Friday

The third fisherman will go fishing on Thursday, Monday, Friday There are 11 days from Monday to Friday the week after

They would all be fishing together again in 11 days

17 А The claims of A and B are identical From the condition of the problem it follows that they are not telling the truth Only C is telling the truth: “A took the golden apple.”

18 3

= 0.232323

As 2016 is an even number, the 2016th digit after the decimal point is

19 20 First way: If we take apples, then the apples left in the first basket would be 16 Then if we take 16 apples from the second basket, apples will be left there The total number of apples left is 20

Second way: If we take x number of apples from the first basket, then the apples left in it would be 18 x Then if we take 18 x from the second basket, there would be 20 (18 x) = + x apples left in it

The total number of apples left is 18 x + + x = 20

20 15 105 = × × = × × ×

SPRING 2016: GROUP

Problem The value of expression is:

A) - 12 084 B) 014 C) 028 D) -2 028

Problem The numerator of the fraction was increased by 20% and the denominator was decreased by 40% The result was a new fraction - The quotient of the two fractions, , is equal to:

A) 0.5 B) 0.75 C) 1.2 D)

Problem Thus reads the postulate of Bertrand: "If n is a positive integer that is greater than 1, then there is always a prime number , where " How many prime numbers are there if = 25?

A) B) C) D) 25

Problem The average age of the crew members excluding the captain is 25 years, and including the captain - 26 years If the captain is 30 years old, then the crew consists of:

A) people B) people C) people D) more than people

Problem The product of the natural numbers from to 122 is divisible by The greatest possible value of the natural number is:

A) 10 B) 11 C) 12 D) 13

Problem The points with coordinates of (0;0), X (2;0), (2;3) and (0;3) are vertices of the rectangle Which one of the following points is outside of the rectangle?

A) A (0;0) B) B (1;1) C) C (2;2) D) D (3;3)

Problem 26 litres of juice must be bottled in 10 bottles of either litre, litres or litres The litre bottles are of an even number How many bottles of litres are there?

A) B) C) D)

Problem If the square is magical, find

A) 11 B) C) 12 D)

Problem What result would you get after calculating the following expression?

Problem 10 When water turns into ice, its volume increases by

If we defrost the ice, how much

would its volume decrease?

A) B) C) D)

Problem 11 If the numbers А and B are such that the expression | | | |, then A + B = …

Problem 12 The number ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ consists of 12 digits (1, 2, 3, , and the number a is included times), and it is divisible by 36 What is the digit a?

Problem 13 The natural number A has natural numbers as its divisors (including and the number itself), the natural number B has natural numbers as its divisors (including and the number itself), and the smallest common denominator of the two numbers is 12 How many natural numbers are there, that are divisors of the number equal to A + B (including and the number itself)?

Problem 14 Three consecutive natural numbers are the digits of hundreds, tens and ones of a three-digit number By how much would this number be increased if we write down its three-digits in reverse order?

Problem 15 Four children met together: Adam, Bobby, Charley and Daniel Adam shook hands with of these children, Bobby shook hands with 2, and Charley shook hands with How many of the children’s hands did David shake?

Problem 16 There are coins, one of which is fake and it is lighter What is the least possible number of times we would have to weigh the coins (using scales) in order to find the fake one?

Problem 17 How many digits are there in the number that is equal to the value of the following expression?

⏟

Problem 18 If the product of integers is a positive number, how many different options are there for the possible number of negative integers among these multipliers?

Problem 19 Three points lie on a straight line The length of each line segment is 2, and k What is the value of k?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C 2 A

3 B The prime numbers that are greater than 25 and smaller than 50 are: 29, 31, 37, 41, 43 and 47

4 B First way: We can check: if the crew members, excluding the captain, are 4, then the sum of their ages would be 100 If we add the captain, then they would be and the sum of their ages would be 130, and 130 5=26

Second way: Let us assume that the crew members, excluding the captain, are equal to The sum of their ages is 25x Including the captain, the sum of their ages would be equal to 26 or 25 + 30 Thus we get the equation 26 = 25 + 30, the solution of which is =

5 C The numbers from to 122 which are divisible by are 61 Among the same numbers, the numbers which are divisible by 11 are 11: 11, 22, .,88, 99, 110, 121 and their product is divisible by because 121 = 11 11 Then the product of the numbers from to 122 can be presented as follows: where A is not divisible by 11, therefore it is not divisible by 22

The greatest possible value of the natural number is 12

6 D The four points lie on the same straight line, and the point D is situated outside the rectangle

7 В Let x, y and z denote the number of bottles containing 1, and liters Then ⏟

We have these options to bottle the juice:

If z = y = x = 3; If z = y = x = 4; If z = y = x = Since the litre bottles are of an even number, only the second option is possible – the number of litre bottles is

The sum of the numbers along one of the diagonals ( 1, 8, Y) is equal to the sum of the numbers along the column (X, 4, Y)

Therefore + = X +( 4) X=11 Such a magical square does exist: 14

20 17

9 C

10 C Let us assume that the volume of the water is equal to kg If it is frozen, it will increase its volume to +

kg The volume of the ice is equal to

kg It would need to be decreased by x, until it reaches Thus we can establish that

Then the volume would be decreased by

11 | | | | | | and | |

12 0 The number must be divisible both by and by The number would be divisible by if = or = The number ̅̅̅̅ would be divisible by if =

13 2 The number is a prime and a divisor of 12 Therefore is either or If = 2, i.e would have more than natural numbers as its divisors If = 3, The number of natural numbers which are divisors of is two and

14 198 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

15 2 A B C D

A + + +

B + – +

C + – –

D + + –

because each hand shake is counted twice In this case the number of hand shakes is + x

We can mark the number of David’s handshakes with x The number x can NOT be greater than

6 + x can be divided by only if x is either or

However, x is not 0, because Adam shook hands with all the children Therefore x = David shook hands with children

16 2 We would have to place coins on each pan of the scales If the scales are balanced:

then we would have to place of the remaining coins on each pan of the scales and if the scales are still balanced, the third remaining coin is fake; if it is not balanced, the lighter (fake) coin can be found on the higher pan of the scales

If the scales are not balanced, the lighter coin can be found on the higher pan of the scales We would then have to compare the weights of two of those three coins

17 1 because the number is found among the multipliers

18 4 The negative numbers among the multipliers are either 0, 2, or There are possibilities

19 5 or Let A, B and C denote the points on the straight line If AB = 2, BC = and

AC = k, the points can be located in possible ways:

А, В, С + = k k = B, A, C + k = k = C, A, B k + = k = C, B, A + = k k = 5

FINAL 2016: GROUP

Problem The product of two integers which are smaller than and greater than ( 8) is equal to The absolute value of the difference of these numbers is ( )

A) B) C) D)

Problem The numbers from to 100 are written down in order: 01234567891011 979899100

If we pick out three consecutive digits, the first two of which have a sum of 10, the third digit cannot be:

A) B) C) D)

Problem What is the remainder when dividing by 12?

A) B) C) D)

Problem Four points have been placed on the square grid below Three of them have the coordinates of ( 5; 0), ( 1; 0) and (0; 0) Find the abscissa of the fourth point

A) B) C) D)

Problem If , the value of the expression is Calculate the value of the expression if х =

A) B) C) D)

Problem Adam has blue, red, white and yellow marbles The number of the blue marbles is more than that of the red ones, the number of the red ones is more than that of the white ones and the number of the white ones is more than that of the yellow ones What is the least possible number of the marbles that Adam has?

Problem A spring with a flow rate of 84 liters of water per minute provides water for three fountains Four times more water reaches the second fountain than does the first, and half as much water reaches the third fountain than does the second By how many liters per minute is the flow rate of the second fountain greater than the flow rate of the third fountain?

A) 12 B) 18 C) 24 D) 30

Problem Three fractions divide the interval between and into four equal parts The greatest of the three fractions is:

A)

B) C) D)

Problem If the square on the right is a magic square, find

X

A) B) - C) D) -2

Problem 10 George chooses numbers The sums of every three of them are 13, 14, 15 and The sum of these numbers is:

A) B) 12 C) 18 D) 42

Problem 11 How many integers А are there, so that = ̅̅̅̅ (where ̅̅̅̅ is a two-digit number) and the two-digit number ̅̅̅̅ can be presented as the product of two different one-digit numbers?

Problem 13 Alex has of each of the following coins: 1, 2, 5, 10, 20 and 50 cents He wants to buy a book that costs euros and 96 cents but he does not have enough money, so he asks his father for help What fraction of the book price does his father need to pay? Write down the answer in the form of an irreducible fraction

Problem 14 Ten teams are participating in a football tournament Each team plays every other team exactly once For each match, the winning team gets points, in the case of a draw both teams get point each, and the losing team gets points, and in case of a draw each team gets point At some point in the tournament it turns out that the teams have earned a total of 131 points How many games are yet to be played?

Problem 15 Calculate , if and

Problem 16 How many of the four-digit numbers that consist of all four digits 2, 0, and 6, are divisible by 36?

Problem 17 A cuboid has been formed from three identical cubes, each with a surface area of sq

cm Calculate the surface area of this cuboid

Problem 18 Five weavers weave 10 rugs in three days How many rugs will weavers weave in days?

Problem 19 For which primes , smaller than 99, is also a prime?

Problem 20 The numbers A and B are such that

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C = ( 2) ( 4), therefore the difference we are looking for is | |

2 А The triplets are

910, 192, 282, 829, 373, 739 , 464, 647, 555, 556, 646, 465, 737, 374, 828, 283, 919, 192 The third digit is never

3 С quotient + remainder It is not difficult to realize that the remainders are among the numbers 0, 2, 4, 6, and 10

It is impossible for the remainder to be or 6, because cannot divide

It is impossible for the remainder to be or 10 either, because can divide , but can not divide 12 quotient + or 12 quotient + 10

There are two remaining options: or

The remainder is 4, because only is divisible by 3, but is not divisible by

4 В The three given points are all on the x-axis

The fourth point has the coordinates of ( 3; 2), so the abscissa is -3 5 D If we replace х with – 3, we would get:

So now the expression becomes Its value is

6 C The number of the white marbles is more than that of the yellow ones; the number of the red marbles is more than that of the white ones, i.e 10 more than that of the yellow ones, and the number of the blue marbles is more than that of the red ones, i.e 12 more than the yellow ones

7 C If the flow rate of the first fountain is , then the flow rates of the second and third respectively are and Concerning we get the following:

We are looking for

The flow rate of the second fountain is 24 liters more than the flow of the third

8 C Have a look at the following fractions:

The fraction we are

looking for is

9 B If we compare the sums of the third row and the second column, we will get the following: 13 + X = 19 + X =

10 B Each of the numbers is part of three sums In order to find the sum of the numbers, we must add the sums and divide the result by We get the following:

( + 13 + 14 + 15) = 12

11 2 Among the perfect squares 16, 25, 36, 49, 64 and 81, only 36 and 81 are such that the numbers 63 and 18 can be presented as the product of two one-digit numbers

Therefore the number A can either be or

12 30 Let us paint the board using a chessboard pattern The cut out squares would be of the same color, and each of the rectangles would cover both colors

In this case it would be impossible to get 31 rectangles However it is possible to get 30 (Two squares would be unnecessary – the ones adjacent to the cut out squares)

13 1/3 Alex paid (1 + + + 10 + 20 + 50) = 264 cents

14 0 or The total number of matches is 45 and the maximum points a team can earn is 135 Now since 131 points have been earned, there are points left

Therefore 131 points can be earned in three ways: 41 wins and draws, with no matches to go; 43 wins and draw, with more match to go; 43 wins, draw and defeat, with no matches to go

15 2139

y – x = 2139

16 6 The sum of the digits is 9, so all the four-digit numbers that consist of these digits are divisible by In order for it to be divisible by 36, it also needs to be divisible by In this case the last two digits must be arranged as follows: ̅̅̅̅̅̅̅ , ̅̅̅̅̅̅̅ , ̅̅̅̅̅̅̅ and ̅̅̅̅̅̅̅ It is now possible to find the four-digit numbers we are looking for: 1260, 2160, 6120, 1620, 2016 and 6012

17 14 The length of the edge of the cube is cm The dimensions of the cuboid are cm ×1 cm ×3 cm , therefore its surface area is

2 × (1×1+1×3+1×3)=14 sq cm

18 14 weaver would weave rugs in days weavers would weave rugs in days weavers would weave rugs in day weavers would weave 14 rugs in days 19 2 If x = 2, then 5x + = 13 is a prime number

If x > 2, then it is an odd number; 5x would also be an odd number, and 5x

+ would be an even number, i.e 5x + would not be a prime number The only prime number is obtained when x =

20 1

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem The sum of the smallest negative integer, which has an absolute value that is smaller than 4, and the smallest positive integer, which has an absolute value that is not smaller than 4, is @ Find @

Problem The segment AB has a length of @ meters The point C separates it in a 1:4 ratio, starting from point A The point D is the midpoint of the segment AC, and the point E is between the points D and B, and divides the segment DВ in a 1:4 ratio The distance from point E to point C is # cm Find #

Problem Less than 80 passengers were traveling on a bus Half of them had occupied the seated spaces # % of all passengers got off at the first stop The number of passengers that got off the bus is & Find &

Problem At least how many numbers we need to choose among all the numbers from (-100) to 100, in order to be sure that at least 18 of them are divisible by & without leaving a remainder? Denote the answer using § Find §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ = The numbers are respectively – and Their sum is

2 # =

𝑐𝑚 𝑐𝑚 𝑐𝑚 𝑐𝑚

3 & =

The number of passengers who got off is of all passengers, i.e a number that is divisible by 25 Out of the the natural numbers smaller than 80, only 25, 50 and 75 are divisible by 25 However, seeing as half of the passengers were seated, then their number must be even

Therefore the number of passengers is 50 Four passengers got off at the first stop

4 § = 168

The numbers from 100 to 100 are 201 Among them, 25 positive numbers, 25 negative numbers, and the number are divisible by A total of 51

In order to make sure that we have chosen at least 18 which are divisible by 4, we would need to choose 201 51 +18 = 168 numbers

5 * =14 Let us reach А in х ways In this case we would be able to reach С in 12

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP

Problem If , then & really represents the operation:

A) + B) C) D)

Problem The value of the expression is:

A) 98,406 B) 9,846 C) D) 60,489

Problem The sum of the absolute values of all integers x satisfying | | and | | > is:

A) B) C) D)

Problem What is the largest natural number n for which

A) B) C) D) 10

Problem If and then is

A) B) – C) D)

Problem Points A, B and C lie on a straight line The distance from A to B is 16 cm The distance from C to A is 10 cm Find the distance from the midpoint of BC to the midpoint of AB.

A) cm B) cm C) сm D) сm

Problem A vessel contains 44 litres of water, while a second vessel contains litres of water The same amount of water has been added to both vessels so that the water in one of the vessels has become times the amount of water in the other vessel How many litres of water has been added to each container?

A) B) C) D) 36

Problem If then equals

А) B) 10 C) 11 D) 12

Problem 10 If the absolute values of the numbers x and y are equal, and , the expression

| | | | | | takes:

А) value B) different values C) different values D) different values Задача 11 If

then the value of is…

Problem 12 Find the value of А, if

Problem 13 In a group of 80 people, 39 have brown hair, 30 have brown eyes and 15 have both brown hair and brown eyes How many have neither brown hair nor brown eyes?

Problem 14 There are X points marked on a circumference, of which are red, and the rest are blue Each two of the points are connected by a segment If the number of segments with two red ends is equal to the number of segments with different-colored ends, how much is X?

Problem 15 The natural numbers are grouped as follows:

What is the sum of the numbers in the tenth group?

Problem 16 I added 100 g mixture of milk and cocoa in a ratio of : to 150 g mixture of milk and cocoa in a ratio of : What is the ratio of the milk and cocoa in the newly received mixture?

Problem 17 What is the three-digit number for which the equality is true

̅̅̅̅̅ ̅̅̅ ̅̅̅ ̅̅̅ ?

Problem 18 How many are the five-digit numbers which end in and are divisible by 3?

Problem 19 Find the smallest prime number, which can be represented as a sum four different prime numbers

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C

2 A 3 D |x|<5 and |x|>3⇒x=4 or x=-4

|4|+|-4|=8

4 B

5 D ⇒

⇒

6 B If the point C is between A and B, then the distance would be 5cm If the point A is between B and C, then the distance would be 5cm Keep in mind that it is NOT possible for B to be between A and C 7 C Carry a check using the given answers: 44 + = 48 and + = 12;

48 12 =

Or use the following solution:

Pour an extra x litres of water Then 44 + x = 4 (8 + x) ⇒ x =

8 A

9 D Other than checking through the given answers, the problem can be solved as follows:

If we mark the number we are looking for with an x, then ⇒ х = 12 Possible answer

10 C The possibilities are –3, –1 and This is how we calculate them: + – = 1; – – = –1; –1 + 1–1 = –1; –1–1–1 =–3

11 ⇒

12 1 First way: we substitute it with x = and conclude that = 10 – 15 + А + – +

Second way:

13 26 With brown hair but not with brown eyes: 39 – 15 = 24; With brown eyes, but not with brown hair: 30 – 15 = 15 With brown eyes and with brown hair: 15

Deduct 24 + 15 + 15 = 54 from the total number in order to get the number of people who have neither brown eyes nor brown hair: 80 – 54 = 26

14 7 The number of segments with two red ends is 10

The number of segments with two different colored ends is 5(X-5) От ⇒

15 1729

Their sum is Another way: 82 + 83 + + 99 + 100 = 1729

The Indian mathematician Ramanujan claims that the number 1729 is the smallest natural number that can be presented, in two

different ways, as the sum of two different numbers raised to the third power

16 11:14 The milk in the first mixture is 30g, and the cacao is 120g In the second mixture the milk is 80g, and the cacao is 20g The total amount of milk is 110g, and the total amount of cacao is 140g The ratio is now 11:14

17 198 ⇒ ⇒

18 3000 The five digit numbers that end in would be divisible by if the four-digit number that we can get after removing the digit of ones of the five digit number is divisible by The total number of four digit numbers are

9999 – 999 = 9000 Each third number is divisible by 19 17 2 + + + = 17

20 16

WINTER 2016: GROUP Problem

A) B) C) D) other

Problem In 1808 the German mathematician Carl Gauss introduced the square bracket notation to denote the largest integer not greater than

What is the value of the following expression: ?

A) -2 B) -1 C) D)

Problem The value of the expression

is equal to:

A) 1,000 B) 10,000 C) 100,000 D) 1,000,000

Problem One of the interior angles of a triangle is 70 degrees, and the difference of two of its interior angles is 30 degrees How many triangles with such angles are there?

(Hint: The sum of the interior angles of a triangle is 180 degrees.)

A) B) C) D)

Problem You can see that points have been placed on the square grid How many acute angles are formed by intersecting the straight lines that connect each pair of points?

(Hint: When two straight lines intersect at a point, they form either acute and obtuse angles or right angles.)

A) B) C) D) another answer

Problem What is the remainder after dividing by 15?

A) B) C) 10 D) 15

Problem The product of two integers that are smaller than and greater than ( 77) is 77 What is the sum of these integers?

A) B) – C) 18 D) 78

Problem When apples are dried they lose 84% of their weight How many kilos of apples would be needed to produce 24 kg of dried apples?

A) 84 B) 100 C) 125 D) 150

A) 10 B) 11 C) 12 D) 13

Problem 10 The code of a safe is made up of all odd digits, without any of the digits repeating What is the maximum number of failed attempts that we would need to make before breaking the code?

A) 120 B) 99 C) 119 D) another answer

Problem 11 You can see a rectangle with a size of 11 cm What is the maximum number of smaller rectangles with sizes of cm we can form out of the big rectangle?

Problem 12 The natural number A would be increased 11 times if, on its right side, we write down one of the following nine digits: 1, 2, 3, 4, 5, 6, 7, or How many digits does the number A have?

Problem 13 In how many different ways can we divide a set of different weights (from to grams each) in two groups of equal weight?

Problem 14 How many proper irreducible fractions are there, whose numerator and denominator are natural numbers with a sum of 33?

Problem 15 What is the smallest natural number N for which the product of 13, 17 and N can be presented as the product of three consecutive natural numbers?

Problem 16 Someone asked Pythagoras about the time, and his reply was as follows:

“The time left until the end of the day is equal to two times two fifths of the time which has already passed” (twenty-four-hour period) What is the time?

Problem 17 If , the value of the expression would be ( 1) What would the value of the expression be if ?

Problem 18 How many prime numbers „ ” are there, for which all three numbers and are primes?

Problem 19 There were 18 apples in one basket, and 20 apples in another basket I took a few apples from the first basket, and then I took as many apples as were left in the first basket, from the second basket How many apples in total are left in both baskets?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B

2 A [–20.15] + [20.15] + [–20.16] + [20.16] = –21 + 20 – 21 + 20 = –2

3 D

4 B Since the sum of the angles of a triangle is equal to two right angles = 180 degrees, the triangles that satisfy the condition of the problem are two and their interior angles are of the following degrees:

70, 40, 70 and 70, 10, 100

5 D Let А, В, С and D denote the points so that D is not on the same line as А, В and С

There are pairs of lines that connect each pair of points: AD and AC, AD and BD, AD and CD, AC and BD, AC and DC, BD and DC

When two straight lines intersect at a point, they form either acute and obtuse angles or right angles AD and BD are perpendicular, hence they form right angles Each of the rest pairs of lines form acute and obtuse angles, hence the total number of acute angles is 10

6 C quotient + remainder

Hence the remainder is a natural number, that is a multiple of 5, but is smaller than 15 i.e it is either 0, or 10

The remainder is not 0, because cannot be divided by

The remainder is not 5, because in this case The number equal to = ⏟

is not divisible by

Therefore the remainder is 10

Among these, two integers are smaller than and greater than 77, hence the numbers we are looking for are and 11 Their sum is 18

8 D 100 % 84 % = 16 % 16% of x is 24 kg

⇒

We need 150 kg of apples to produce 24 kg dried apples

9 C А = 2n + + 2n + + 2n + = 6n + 9,

therefore B = 2n – + 2n – + 2n + = 6n – or B = 2n + + 2n + + 2n + = 6n + 21 Therefore A – B = 12 or A – B = –12

10 C All possible codes are = 120 In the worst case scenario, we would break the code after making 119 failed attempts The code would be the 120th possible number

Therefore we would break the code after the 119th attempt

11 D 99 6=16 + a remainder of 4, therefore 16 is most likely to be the greatest number

12 1 If we write down the number ̅̅̅̅ where x is one of the digits from to 9, then 10A + x = 11A Then x = A, i.e A is a one-digit number

13 4 The total weight of the set of weights is 28 grams If the weight of grams is in the first group, then we would need to add other weights in the group that have a total weight of grams = + = + = + = + + 4, therefore we could group them in ways

First way: first group – 7, 6, 1; second group – 2, 3, 4,

Second way: first group – 7, 5, 2; second group – 1, 3, 4,

Third way: first group – 7, 4, 3; second group – 1, 2, 5,

14 10 First way: We write down all proper fractions with 33 as the sum of their numerator and denominator and then we remove all reducible fractions: 1/32; 2/31, 4/29, 5/28; 7/26, 8/25; 10/23; 13/30; 14/19; 16/17

Second way: The numerators of the fractions we are looking for would be all natural numbers, smaller than 17 and non divisible by divisors of 33 other than and 33, i.e and 11

15 600 13 17 = 52 51, therefore the smallest natural number we are looking for is 50 = 600

13 17 600 = 50 51 16 13 h 20

min

The time that has already passed can be denoted with x The time left can be denoted with Therefore the equation can be presented as

⇒ ⇒ 17 therefore A =

Then if x = 2, the value of the expression would be

18 1 If p is a prime number, then p = 3, or when divided by leaves a remainder of or

If p = 3, all three numbers are prime: 3, 14, 17

If, when divided by 3, the number p leaves a remainder of 1, then the number

p+14 is divisible by itself, by 1, and by 3, i.e it is not a prime number

If, when divided by 3, the number p leaves a remainder of 2, then the number

p+10 is divisible by itself, by 1, and by 3, i.e it is not a prime number

Therefore there is only one number (p = 3) for which all three numbers and are primes

19 20 First way: If we take apples, then the apples left in the first basket would be 16 Then if we take 16 apples from the second basket, apples will be left there The total number of apples left is 20

Second way: If we take x number of apples from the first basket, then the apples left in it would be 18 x Then if we take 18 x from the second basket, there would be 20 (18 x) = + x apples left in it

The total number of apples left is 18 x + + x = 20

SPRING 2016: GROUP

Problem If , then the value of the expression | | | | is:

A) +1 B) C) D)

Problem What would need to be equal to, in order to get the smallest possible value of the following expression ?

A) 2016 B) C) D)

Problem The sum of the two-digit numbers ̅̅̅ and ̅̅̅ cannot be …

A) 66 B) 154 C) 198 D) 155

Problem The value of the expression

can be calculated after simplifying the expression The value of is:

A) 2017 B) 2016 C) 2015 D) other

Problem All possible values of the parameter , for which is the root of the equation , are the numbers:

A) B) C) and D) and

Problem To what power we need to raise in order to get ?

A) B) C) 12 D) 24

Problem If and , then is:

A) a non-negative number B) a negative number

C) a positive number D) cannot be determined

Problem If each of the angles of a quadrangle is the average of the other three angles, then this quadrangle would always be:

A) a parallelogram B) a rhombus C) a square D) a rectangle

Problem How many of the solutions of the equation are solutions of the inequality | |

A) B) C) D)

Problem 10 After a price increase of 20 % an item cost $ 240 The price of the same item before the price increase was:

Problem 11 The numbers 0, 1, 2, 3, and have been used to write down all four-digit numbers that have no repeated digits and are divisible by What part of these numbers are divisible by 10?

Problem 12 26 litres of juice must be bottled into 10 bottles of either litre, litres or litres The litre bottles are of an even number How many bottles of litres are there?

Problem 13 If | | | | where is a positive integer, what is the smallest value of

Problem 14 If is a prime number, how many possible remainders would we get when dividing

p by 6?

Problem 15 The polynomial is expressed as follows: What is the value of ?

Problem 16 On each side of a cube with an edge of cm is glued a cube equal in size Find the surface area of the resulting body in cm2

Problem 17 12– 22 + 32– 42 + 52– 62 + + 992 −1002 = …

Problem 18 In a chess tournament, each contestant must play a game with each of the other contestants There are contestants in the tournament: Alexander, Boris, Carl and David So far Alexander has played games, Boris has played games and Carl has played games How many games has David played?

Problem 19 If the product of 100 numbers is a negative number, how many different options are there for the possible number of negative integers among these multipliers?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 D ⇒ | | For each | |

Therefore:

| | | |

2 D =

We can get an equality when both and We get that

3 D ̅̅̅ ̅̅̅ Therefore the sum of ̅̅̅ ̅̅̅ is always divisible by 11 Among the given numbers 154 is not divisible by 11

4 D ⇒

5 D If is a root, then е ⇒ ⇒ ⇒ ⇒ or

6 C Let denote the number we are looking for

⇒ ⇒ ⇒ ⇒

7 B and ⇒

and ⇒

and ⇒ ⇒ ⇒

⇒ ⇒

In the same way we can get that

Thus we reach the conclusion that the quadrangle is a rectangle

9 C The numbers 2, and – are solutions of the equation

Among them and – satisfy the inequality | |

10 A Let denote the price we are looking for In this case

⇒

11 5/9 The numbers divisible by are either of the ̅̅̅̅̅̅̅ kind or of the

̅̅̅̅̅̅̅ The numbers of the ̅̅̅̅̅̅̅ kind are divisible by 10

The numbers of the ̅̅̅̅̅̅̅ kind are = 60, and the numbers of the

̅̅̅̅̅̅̅ kind are 4 = 48

The part we are looking for is 60 108 = 5/9

12 Let x, y and z denote the number of bottles containing 1, and liters Then ⇒ ⏟

⇒ ⇒ We have these options to bottle the juice: If z = ⇒ y = ⇒ x = 3;

If z = ⇒ y = ⇒ x = 4; If z = ⇒ y = ⇒ x =

Since the litre bottles are of an even number, only the second option is possible – the number of litre bottles is

13 If N = ⇒ | | | | If N = ⇒ | | | |

If ⇒ | | | |

possibilities would be and However in and the number p is not a prime Then the possible remainders are and

Now we must add the remainders and 3, which we get when dividing the primes and by

If ⇒ the possible remainders are and 5; If ⇒ the possible remainders are and There are possible remainders in total

15 6 The identity = is also true for Therefore

= ⇒ 16 30 The number of cubes is The resulting body is made up of faces of

each of the cubes = 30 faces in total, each one with an area of sq.cm = 30 sq.cm

17 – – – –

= ⏟

18 1 or If we add the number of games each contestant has played, the number we get must be divisible by 2, because each game is counted twice

In this case the number of games is + x, where x denote the number of games David has played The number x cannot be greater than

There are four possibilities: 0, 1, and 7 + x can be divisible by if x = or x =

19 50 The negative numbers must be an odd number The possibilities are 1, 3, 5, , 99 50 in total

20 2 Let denote the price of an apple The price of a pear would then be =

From the second condition we get that the price of a pear is equal to

We get the equality ⇒ Then = 0.5

FINAL 2016: GROUP

Problem The product of 100 integers is 100 What is the smallest possible sum of these numbers?

A) -199 B) -195 C) -2 D)

Problem The numbers from to 100 have been written down in order: 01234567891011 979899100 If we were to erase three consecutive digits, the first two of which have a sum of 10, the third digit will most often be:

A) B) C) D)

Problem 3. The sum of the two-digit numbers ̅̅̅ and ̅̅̅ is a square of a natural number and is equal to:

A) 196 B) 169 C) 144 D) 121

Problem What remainder is left after dividing by 12?

A) B) C) D)

Problem If the following is true for each value of a:

then = ?

A) B) C) D)

Problem At least how many composite integers, smaller than 50, we need to select, so that at least two of them would have a common divisor that is greater than 1?

A) B) C) D)

Problem The number (-1) is the root of the equation

where is the unknown and A is a parameter In this case A =

A) -1 B) C) D)

Problem The point M is placed on the side BC of the triangle ABC so that CM = MA = AB and AC = BC In this case =

A) B) 1.5 C) 2.5 D)

Problem

Problem 10 We have a square with a side of 10 cm We have cut out smaller squares, each with a side of 1cm, from two opposite corners of the big square What is the greatest possible number of rectangles with sizes of 1cm by cm that we can divide the newly formed figure into?

A) 98 B) 49 C) 48 D) different answer

Problem 11 Determine all integers A, such that = ̅̅̅̅ and the two-digit number ̅̅̅̅ can be presented as the product of two consecutive odd numbers?

Problem 12 The water content in kg of fresh mushrooms is 90% After drying the mushrooms, the water content is now 20% of their weight How many grams the dried mushrooms weigh? Problem 13 In a particular year there are three consecutive months with Sundays in each What are the possible sums of the numbers of days in those three consecutive months?

Problem 14 Let us have a look at the following pairs of numbers:

Find n, if the sum of the digits of the numbers of each pair is 11

Problem 15 An isosceles triangle with a leg of cm and a square with a side of cm have equal areas How many degrees is the smallest angle of the triangle?

Problem 16 If N is an integer, how many are the possible remainders when dividing by 5? Problem 17 After having covered of my journey, plus another 200 m, I have yet to go another 50 m less than the distance I have already covered How many km is the length of my journey?

Problem 18 In 1808, the German mathematician Carl Gauss introduced the indication He used it to denote the greatest integer that is not greater than

Calculate the value of the expression [[ ]

] [ ]

Problem 19 Five weavers weave 10 rugs in days How many rugs would weavers weave in days?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 A

100 = ( 100) ⏟

⇒

the sum we are looking for is ( 199)

2 С The digit triplets are 910, 192, 282, 829, 373, 738 , 464, 647, 555, 556, 646, 465, 737, 374, 828, 283, 919, 192 The third digit is most often

3 D

̅̅̅ ̅̅̅ ⇒

⇒ ̅̅̅ ̅̅̅

4 С

, therefore it is not hard to conclude

that the remainders are among the numbers 0, 2, 4, 6, and 10 It is not possible for the remainder to be or 6, because can not divide It is not possible for the remainder to be 2, because divides but can not divide 12 quotient +

There are two possibilities left: or

The remainder is 4, because only is divisible by 3, but is not divisible by

5 D

therefore

=

6 В

Each composite integer that is smaller than 50 is divisible by one of the prime numbers 2, 3, and

Therefore according to the principle of Dirichlet, it would be enough to choose numbers in order to be sure that among them there would be two which have a common divisor greater than

7 D ⇒ ⇒

8 A ⇒ ⇒

9 A

10 C

Let us paint the board using a chess pattern The squares which have been cut out are of the same color, and each of the rectangles covers both colors In this case it would not be possible to get 49 rectangles However it is possible to get 48

(Two of the squares can not be used to form a rectangle – the ones adjacent to the squares which have been cut out)

11 2

Out of the perfect squares 16, 25, 36, 49, 64 and 81, only 36 satisfies the initial condition: 63 can be presented as the product of two consecutive odd numbers

In this case the number A can either be or -6

12 625

Let us denote the weight of the mushrooms in kilograms using x We can compare the weight of the “dry” material:

⇒

13 89 or 90

The possibilities are:

31+28+31=90 или 31+29+31=91; 28+31+30=89; 29 + 31 + 30=90 31+30+31=92; 30+31+30=91; 31+30+31=92; 30+31+31=92; 31+31+30=92; 31+30+31=92; 30+31+30=91; 31+30+31=92

Between 12 Sundays there are 12 + 11 = 78 days;

Therefore from the sums we will have the following options left: 12 days or 13 days; 11 or 12; 14; 13; 14; 14; 14; 14; 13; 14 February is definitely among those months

The possibilities are:

14 28

The numbers smaller than 28 not satisfy the condition of the problem For example, is not possible, one of the groups will have the numbers 19 and 9, and in this case the sum of the digits would be 19 The numbers greater than 29 not satisfy the condition of the problem either, because there will be a group that has the number 29 and the number However the sum of the digits of the number is a number greater than In this case in the group of numbers the sum of the digits would be greater than 11

15 15 or 30

If the triangle has an acute angle between its legs, then it would be In this case, the smallest angle would be - that would be the angle between the legs

If the triangle has an obtuse angle between its legs then it would be In this case, the smallest angle would be – these would be the angles at the base

16 2 The fourth power of any natural number has 0, 1, or as its digit of ones In this case when divided by the remainder would be either or

17

⇒

Let us denote my full journey with X m In this case ⇒

18 [[ ]] [

]

19 14

1 weaver will weave rugs in days; weavers will weave rugs in days; weavers will weave rugs in day; weavers will weave 14 rugs in days

TEAM COMPETITION – NESSEBAR, BULGARIA MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem is an irreducible fraction with natural numbers for a numerator and a denominator, such that and The number of all such fractions is @ Find @

Problem The number @ has shown up on a computer screen After each hour that passes, the number gets replaced by the sum of the number and the product of its digits Which number will show up on the screen after 2016 hours? Denote the answer using # Find #

(If the number is 11, the number that shows up on the screen after the first hour will be equal to

11 + 1 = 12, then the number that shows up on the screen after the second hour will be equal to

12 + = 14, )

Problem Two squares have sides that are measured in integer centimeters The sum of the areas of the squares is (# + 68) The greatest possible perimeter of the figure is & cm Find &

Problem The product of & consecutive two-digit numbers is divisible by The smallest number in this product is § Find §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ = 24

The numerator a can be any integer from to 52 In order for the fraction to be irreducible, we must eliminate all of the possible numerators that are divisible by 3, and (the prime divisors of 105)

The numbers divisible by are 17 The numbers divisible by are 10 The numbers divisible by are

The numbers divisible by 3, and by 5, are The numbers divisible by 3, and by 7, are The number divisible by 5, and by 7, is

Therefore the numbers that are divisible either by 3, or by 5, or by 7, are 17 + 10 + – (3 + + 1) = 28

The irreducible fractions are 52 – 28 = 24

2 # = 170

The numbers that will show up on the screen are: 24; 24 + = 32; 32 + = 38; 38 + = 72;

72 + = 86; 86 + = 134; 134 + = 146; 146 + = 170; 170 + = 170; 170; 170;

3 & = 58

If the lengths x and y are the lengths of the sides of the squares, the perimeter of the figure is 4x + 2y, x > y

4 § = 18

The product of the numbers from to 67 contains

[ ] [ ] numbers when factorized as a product of simple multipliers

The product of the numbers from to contains

[ ] [ ] number when factorized as a product of simple multipliers

In this case the product of the numbers from 10 to 67 contains 15 - = 14 numbers when factorized as a product of simple multipliers

10 11 66 67 contains 14 multipliers 5; 11 12 … 67 68 contains 13 multipliers 5; 12 13 … 68 69 contains 13 multipliers 5; 13 14 69 70 contains 14 multipliers 5; 14 15 70 71 contains 14 multipliers 5; 15 16 71 72 contains 14 multipliers 5; 16 17 … 72 73 contains 13 multipliers 5; 17 18 … 73 74 contains 13 multipliers 5; 18 19 74 75 contains 15 multipliers The smallest number we are looking for is 18

5 * =

Let the sides be denoted as a, b and c and let In this case ⇒

From the inequality of the sides of the triangle ⇒ ⇒ we can find the possible values of c: cm, cm and cm If ⇒

If ⇒

If ⇒

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP 8 Задача If then the value of is:

A) 2,015 B) C) 4,030 D)

Problem When the natural number is divided by 6, the remainder is When the natural number is divided by 6, the remander is What is the remainder when is divided by 6?

A) 0 B) C) D)

Задача If and , then | |

A) 3 B) C) D)

Problem Find the tens digit in the value of

⏟

A) 0 B) C) D)

Problem If ab>0 and a+b<0, then the value of | | | | is:

А) B) C) D)

Problem A polygon has more than 40 diagonals The number of its sides is at least:

А) B) 10 C) 11 D) 12

Problem When solving the same quadratic equation, three students got different roots: The first student got the numbers and as roots;

The second student - and 3; The third student - and

It turned out that each student got exactly one root of the equation right If и are the roots of the equation, then – is:

A) B) C) 4 D) 9

Задача If √ and √ , how many of the numbers and are rational?

Problem The numbers a and b are such that the expression has the smallest value possible The value is:

A) B) 1 C) D)

Problem 10 A rectangle is divided by two intersecting lines parallel to its sides Four smaller rectangles are formed, three of which have areas of sq.cm, sq.cm and sq.cm Find the smallest possible area of the fourth rectangle

А) B) C) D)

Problem 11 What is the largest possible number of acute angles in a convex hexagon? Problem 12. How many are the five-digit numbers which end in and are divisible by 3?

Problem 13. How many are the integers smaller than 2015, which can be represented as a sum of two consecutive integers and as a sum of three consecutive integers?

Problem 14 For how many integers , the value of is a prime number?

Problem 15. What is the greatest number of cells 1x1 you can paint in a 11x11 square drawn on graph paper, so that no one box 2x2 did not have three of painted cells?

Problem 16. For how many integers , the numbers

and

are also integers?

Problem 17. In how many ways can we distribute identical pears between children so that each child receives at least one pear?

Problem 18 Find the integer , if √ √ √

Problem 19. The point D is of the median CM of the triangle ABC and it is such that СD=DM If the point E is an intersection of the straight line AD and the side BC, determine CE:CB

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C ⇒ X=8060, Y=4030⇒X-Y=4030

2 A

3 A ⇒ ⇒ | |

4 A After we cancel the fraction, we would get a number that is the product of a number and 100 The digit of tens is

⏟

5 B ⇒ | | | | ⇒

⇒ | | | |

6 C The number of diagonals of an N-triangle can be found using the formula In this case, if we carry out a check, we will find that:

Therefore, the polygon would have 11 angles

7 C The roots are either and 3, or and Therefore the value we are looking for is

8 C From √ и √ , it follows that only and are rational numbers

10 D If we mark the areas of the rectangles with and ,

Then the smallest area would be

11 3 The sum of the angles of a convex hexagon is 720◦ and each of them is smaller than 180◦ If of the angles are acute, then the sum of the other two would be greater than 360◦, i.e at least one of them would be greater than 180◦

12 3000 The five digit numbers ending in are divisible by if:

The four digit number that we got after removing the digit of ones from the five digit number, is divisible by The total number of four digit numbers is 9999-999=9000 Each third number is divisible by 13 336 The sum of two consecutive natural numbers is odd, and the sum of

three consecutive natural numbers is divisible by

The odd numbers that are smaller than 2015 and are divisible by are 3, 9, 15, , 2013 Their total number is 336

14 3 The numbers

17-0, 17-4 are the primes for three of the values of n=3, 5,

15 66 If we paint all odd columns, the number of colored squares would be

⏟

16 0

;

⇒ ⇒ ,

which is impossible

17 15 First solution: 7=5+1+1=4+2+1=3+3+1=3+2+2, therefore the

following positioning can be done in: 5, 1, - ways;

4, 2, – ways; 3, 3, – ways 3, 2, – ways 15 ways in total

Second solution: Let us add two identical apples to the pears and

arrange the pieces of fruit in a row Let us now arrange the pears as follows:

On the second row – from the first to the second apple; On the third row – from the second apple to the end

We can place the first apple in the second place, in which case the places available for the second apple would be to 8, i.e possibilities

We can place the first apple in the third place, in which case the places available for the second apple would be to 8, i.e possibilities

We can place the first apple in the fourth place, in which case the places available for the second apple would be to 8, i.e possibilities

We can place the first apple in the fifth place, in which case the places available for the second apple would be to 8, i.e possibilities We can place the first apple in the sixth place, in which case there would only be one place left available for the second apple, that is, 8, i.e possibility

5+4+3+2+1=15 possibilities in total

18 2 First solution: √ √ ⇒ √

√ ⇒ ⇒

Second solution: Apply the following formula

√ √ √ √ √ √

19 1:3 ⇒ ⇒

M is the midpoint of AB Therefore the areas of and are equal ⇒

Since the areas of the triangles ACM and BCM are equal, it follows that 2F=3S+F⇒F=3S⇒ 4S=1:2

⇒

WINTER 2016: GROUP

Problem If the equation is an identity, then

A) B) C) D) other

Problem The square of the natural number A is recorded with the digits 0, 2, and Which digits would we use to record A?

A) 0, 2, B) 0, 2, C) 2, and D)another answer

Problem If , then | |

A) 17 B) 33 C) 65 D) 129

Problem One of the interior angles of a triangle is 70 degrees, and the difference of two of the interior angles of this same triangle is 30 degrees How many such triangles are there?

(Hint: The sum of the interior angles of a triangle is 180 degrees.)

A) B) C) D)

Problem 5 You can see that points have been placed on the square grid How many obtuse angles will be formed by intersecting the straight lines that connect each pair of points?

(Hint: When two straight lines intersect at a point, they form either acute and obtuse angles or right angles.)

Problem It is known that the sum of more than consecutive natural numbers is 20 How many possibilities are there?

A) B) C) D)

Problem Find the area (in sq cm) of an isosceles triangle with an angle of and a 10 cm long leg

A) 100 B) 50 C) 25 D) 12.5

Problem Someone asked Pythagoras about the time, and his reply was as follows:

“The time left until the end of the day is equal to two times two fifths of the time which has already passed” (twenty-four-hour period) What is the time?

A) 13 h 20 B) 13 h 40 C) 14 h 20 D) 14 h 40

Problem Peter added consecutive odd numbers and the sum he got as a result was A Steven added consecutive odd numbers and the sum he got as a result was B If one of Peter’s numbers is the same as one of Steven’s numbers, then the greatest possible difference of the sums they both got (A and B), is:

A) 10 B) 11 C) 12 D) 13

Problem 10. In a rectangular triangle, and are the legs, is the hypotenuse, is the height to the hypotenuse Which of the following sums is the greatest?

A) B) C) D)

Problem 11 You can see a rectangle with a size of What is the maximum number of smaller rectangles with sizes of we can form out of the big rectangle?

Problem 13 The isosceles triangles and have been built on the sides of the square If , then calculate

Problem 14 How many proper irreducible fractions are there, whose numerator and denominator are natural numbers with a sum of 41?

Problem 15 What is the smallest natural number N, for which the product of 13, 17 and N can be presented as the product of three consecutive natural numbers?

Problem 16 Calculate √ √ √ √

Problem 17 Find the perimeter of the quadrilateral that we would get when connecting the midpoints of the sides of a quadrilateral with diagonals equal to cm and cm

Problem 18. Find the sum of the coefficients of the even degrees in the simple form of the polynomial

Problem 19. What is the greatest possible value of the number N that would make the following statement true: “N numbers can always be found among 97 random integers, in a way that the difference of each pair would be divisible by 8.”?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C First way:

Hence = 125 + ( 300) + 240 + ( 64) =

Second way:

If x = 1, then ⇒ 2 B The square of a natural number can not end neither in 2, nor in If it ends in

0, that means that it would end in two zeros Therefore the square of number

A ends in The possibilities are 3204; 3024; 2304 and 2034

We would have to exclude 2034 as a possibility because A is an even number and the square of A is divisible by 4, whereas 2034 is not divisible by From 3204 = 89, 3024 = 84; 2304 = 64 it follows that

number A is 48, and 48 = 240

3 A , therefore

√ , i.e | |

4 C Since the sum of the angles of a triangle is equal to two right angles = 180 degrees, the triangles that satisfy the condition of the problem are two and their interior angles are of the following degrees:

70, 40, 70 and 70, 10, 100

5 D Let А, В, С and D denote the points so that D is not on the same line as А, В and С

There are pairs of lines that connect each pair of points: AD and AC, AD and

BD, AD and CD, AC and BD, AC and DC, BD and DC

When two straight lines intersect at a point, they form either acute and obtuse angles or right angles AD and BD are perpendicular, hence they form right angles Each of the rest pairs of lines form acute and obtuse angles, hence the total number of obtuse angles is 10

6 В There is possibility: + + + + 20

Therefore the height to the leg would be cm and the area would be 25 sq.cm

АВС is an isosceles triangle ⇒ АС = ВС

Let АН⏊BC (H∊BC) ⇒Δ ACH, HCA= ⇒ ⇒ the area of = 25 sq cm

8 A The time that has already passed can be denoted with x The time left can be denoted with Therefore the equation can be presented as

⇒ ⇒ 9 C

therefore – – – or Therefore – или – –

10 D From and

it follows that is the greatest sum

11 16 99 = 16 + a remainder of 4, therefore 16 is most likely to be the greatest number

12 240 Let us mark the two students as student A and student B We can arrange them and the other four students in = 120 ways In each of these ways the students A and B can be arranged either as AB or as BA Therefore they can be arranged in a total of 120 = 240 ways 13 is equilateral, therefore ANM =

is isosceles ⇒ ⇒

ΔABM is isosceles with legs AM and BM and MAB =

Therefore Δ ABM is equilateral and the measure we are looking for is 60 degrees

14 20 First way: We write down all proper fractions with 41 as a sum of their numerator and denominator and then we remove all reducible fractions, if there are any All proper fractions with 41 as a sum of their numerator and denominator are irreducible: 1/40, 2/39, 3/38, , 20/21

than and 41 The possible numerators are all natural numbers from to 20 15 600 13 17 = 52 51, therefore the smallest natural number we are

looking for is 50 = 600 13 17 600 = 50 51

16

√ √ √ √ √ √ √ √

| √ | | √ | ( √ ) √

17 9 cm The quadrangle we get as a result is а parallelogram with sides equal to the halves of the diagonals, i.e 2cm; 2.5cm; 2cm and 2.5cm The perimeter is

cm.

18 1

⇒

⇒

19 13 The difference of two numbers would be divisible by if they both leave the same remainder when divided by

The number of remainders after a division by is 8: 0, 1, 2, 3, 4, 5, and Among the 97 numbers that are not divisible by 8:

12 leave a remainder of 0; 12 leave a remainder of 1, 12 leave a remainder of , ……… , 12 leave a remainder of

12 therefore there is number left This number will also leave a remainder which will be a number between and

Therefore the greatest number of pairs of numbers among 97, with a difference divisible by is 13

SPRING 2016: GROUP 8

Problem 1. The greatest negative integer that is a solution of the inequality | | √ is:

A) B) C) D)

Problem The sum of three numbers is 222 If we increase the first number by 2, increase the second number twice, and diminish the third number twice, we would get the same number Find the smallest of the three numbers

A) 22 B) 32 C) 42 D) other

Problem 3. What is the value of the following expression? √ √ √

A) √ B) √ C) D)

Problem 4. What power we need to raise to, in order to get ?

A) B) C) 12 D) 24

Problem There are points on the circumference of a circle What is the greatest possible number of right-angled triangles that have these points as their vertices?

A) 24 B) 30 C) 36 D)

Problem 6. Two years ago A was twice older than B, and three years ago B was three times younger than A How old is A now?

A) 12 B) 10 C) D)

Problem For how many of the integers n can we claim that is divisible by ?

A) 0 B) C) D) more than

Problem What remainder is left when is divided by 13?

A) B) C) D)

Problem 26 litres of juice must be bottled into 10 bottles of either litre, litres or litres In how many ways can we this, if we use all three bottle sizes?

A) B) C) D)

Problem 10 In the graph | |, where is the parameter, and the coordinate axes determine a triangle with an area of What is the smallest possible value of the expression ?

Problem 11 If N and M are natural numbers such that √ √ , calculate

Problem 12 (inspired by a problem by Johannes Buteo, who lived during the 16th century) The price of apples, minus the price of a pear, is equal to $13, and the price of 15 pears, minus the price of an apple, is equal to $ How much would one apple and one pear cost?

Problem 13 The diagonals of a trapezium divide it into four triangles Three of the areas of the triangles are equal to respectively 4, and sq.cm What is the area of the trapezium?

Problem 14 What is the number of real roots of the equation | | ?

Problem 15 If A and B are 4-digit natural numbers, how many solutions can you find to the following equation ?

Problem 16. The numbers 187 and 219 leave the same remainder (11) when divided by the number Find the number

Problem 17 Four children: A, B, C and D must be arranged in a row, in such a way that A and B, and

C and D, would always be standing next to each other In how many different ways can we this?

Problem 18 Calculate the sum of the reciprocal and the opposite of number if √

Problem 19. If each of the angles of a quadrangle is the average of the remaining three angles, find the largest angle

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B Check each possible answer, starting with and until you reach the answer, that is

2 B If denote the number that we get after increasing the first number by 2, then the first number would be , the second number would be , and the third number would be As a result, we would get the

following equation:

The first number is 62, the second number is 32 and the third number is 128

3 C

√ √ ( √ ) | √ | ( √ ) (√ ) ( √ )

4 B If denote the number we are looking for ⇒ ⇒

⇒ ⇒ ⇒

5 А If we place the points two by two, so that they would be the edges of a diameter, we would get right-angled triangles with a common hypotenuse for each diameter There are right-angled triangles in total

6 D А B

Three years ago

Two years ago

The eqation is

⇒

Now

7 B

and therefore

is an integer if

8 D

The expression is divisible by 13, and the remainder would be

9 B Let and denote the number of bottles of l and l respectively Then the bottles of litres would be – –

⇒ By using we can fill in the following table:

1 l 3 l 5 l

6 bottles 0 bottles 4 bottles

5 bottles 2 bottles 3 bottles

4 bottles 4 bottles 2 bottles

3 bottles 6 bottles 1 bottles

2 bottles 8 bottles 0 bottles

From the table above we can see that the number we are looking for is

10 B In the graph of | |, where is the parameter, and the coordinate axes determine the triangle

with an area of ⇒

In this case the smallest possible value of the expression is

11 3 √ √ √

If ⇒ √ is an irrational number, then Therefore M=1 and M+N=3

12 2 Let x denote the price of an apple The price of a pear would then be equal to - 13

From the second condition we get that the price of a pear is equal to

We get the equality

⇒

Then = 0.5

13 25 ABCD is the trapezium, O is the intersection of its diagonals, AB The areas of the triangles ADO and BCO are equal, therefore the possible areas of the four triangles are 4, 4, 6, 9; 4, 6, 6, 9; 4, 6, 9,9

and and ,

therefore the areas of the triangles are 4, 6, and

In this case the area of the trapezium is + + + = 25

14 2 | | ⇒

, if The root is equal to , if The root is equal to

15 6984 , therefore the value of A is the smallest, if B The value of A is the greatest, if

The numbers are 7983 – 999 = 6984

16 16 We are looking for a natural number greater than 11, which is a divisor of both 187 – 11 and 219 – 11 This number is 16

17 8 We must arrange the X and Y pairs, which are respectively made up of

A and B, C and D

X and Y can be arranged in ways, and each of X and Y can also be arranged in ways 2 = ways in total

18 2

√ ⇒ √ √ ⇒

19 90 The sum of the angles of every quadrangle is 360 degrees Let the angles

⇒ ⇒ ⇒ ⇒ In the same way we can reach the conclusion that t

20 30 The identity = is also true for Therefore

FINAL 2016: GROUP

Problem 1. If the number a is rational and the number √ is also rational, then the smallest value of b is:

A) B) C) 2 D)

Problem 2. Take a look at the following pairs of natural numbers If the sum of the digits of the numbers from each pair is 23, find n

A) 499 B) 994 C) 949 D) different answer

Problem 3. In the numerical equality known as „the problem of the Indian mathematician Bhaskara” the last number is replaced with the letter A Find A

√ √ √ √ √ √

A) B) C) √ D) √

Problem 4. A straight line has been built through the vertex A of the parallelogram ABCD and the point M of the diagonal BD and this straight line divides the diagonal BD in a 1:2 ratio starting from the vertex D In what ratio does this straight line divide the side CD starting from the vertex D?

A) 1:1 B) 1:2 C) 2:1 D) 3:1

Problem If the following is true for each value of a,

then

A) - B) -1 C) D)

Problem What is the sum of the real roots of the following equation?

A) -1 B) C) D)

Problem 7. Find the distance from the intersection point of the graphs of the functions and to the ordinate axis

Problem A square and a circle have a common part The area of the square, the area of the common part and the area of the circle are relative to each other 4:1:17 What percentage of the figure below is the area of the common part?

A) B) 10 C) 15 D) 20

Problem 9. In 1808, the German mathematician Carl Gauss introduced the indication [ ] He used it to denote the greatest integer that is not greater than How many natural numbers n are there, for which [ ] is a prime?

A) B) C) D) more than

Problem 10 How many points (x, y) are there that have negative integers as coordinates and ?

A) B) C) D) more than

Problem 11. is an equilateral triangle with sides of length cm The points M, N and P are respectively found on the sides BA, AC and CB, and are such that MN⏊AC, NP⏊CB and PM⏊AB Calculate the length of the segment AM

Problem 12 Calculate the difference of the real numbers and , if and

Problem 14. What is the smallest possible number of different digits that we can use to write down numbers, which when divided by leave different remainders?

Problem 15 In a particular year three consecutive months have Sundays each What are the possible sums of the days of these three consecutive months?

Problem 16 Calculate the value of the following expression:

√ √ √ √ √ √ √ √

Problem 17. We have a square with a side of 10 cm We have cut out smaller squares, each with a side of 1cm, from two opposite corners of the big square What is the greatest possible number of rectangles with sizes of 1cm by cm that we can divide the newly formed figure into?

Problem 18 Calculate the product of the real roots of the following equation:

Problem 19. If N is an integer, how many possible remainders are there when dividing by 5?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 A

The number √ would be a rational number if We get two possible values for : and (-1)

In this case the smallest value of is

2 D n = 598

3 C

√ √ √ √ √ √ ⇒

√ √ √ √ ⇒ √ √ √ √ ⇒ ( √ ) √ √ √ =0⇒ √

4 A

In a triangle ACD the median is DO (O is the intersection of the diagonals of the parallelogram) However in this case the point M would be a centroid, which means that AM will intersect the side CD in its midpoint

5 A

=

i.e

6 А

First we need to note that is not a root of the equation If ⇒ ( )

⇒

We can now come to the conclusion that the roots are and (-1) Their sum is (-1)

7 B The intersecting point of the two graphs is a point with coordinates of (3;- 6) The distance we are looking for is

8 A

The areas of the square, the common part, and the circle are respectively and The surface of the whole figure is 20k The percentage we are looking for is

[ ] We get a composite number, because if m = 1, then

If , where m is a natural number or 0, then [ ]

We get a composite number because и If , where m is a natural number or 0, then

[ ] We will only get prime numbers for i.e for

10 C The inequality is true only for

11 2

If ⇒

From and the above calculations we can reach the following equation

Such a triangle does exist

12 or (- 5)

We can conclude that and In this case ⇒ | |

The difference between the two numbers is either or (-5) There are such real numbers

13 77 283 The greatest product is 76421 853

The sum of the multipliers is 76421 + 853 + 9= 77 283

14

For example:

12, 1, 2, 111, 22, 111, when divided by 6, leave the following

remainders: 0, 1, 2, 3, и They can be written down with digits

15 89 or 90

The possibilities are:

31+28+31=90 или 31+29+31=91; 28+31+30=89; 29 + 31 + 30=90 31+30+31=92;

30+31+31=92; 31+31+30=92; 31+30+31=92; 30+31+30=91; 31+30+31=92

Between 12 Sundays there are 12 + 11 = 78 days;

Therefore from the sums we will have the following options left: 12 days or 13 days; 11 or 12; 14; 13; 14; 14; 14; 14; 13; 14 February is definitely among those months

The possibilities are: 31+28+31=90; 28+31+30=89; 29 + 31 + 30=90

16 √ √ √ √ √

√ √ √ √ √ √ √

17 48

Let us color the board using a chessboard pattern The squares which have been cut out are of the same color, and each of the rectangles covers both colors

In this case it would be impossible to get 49 rectangles However it is possible to get 48 rectangles

(Two of the squares can not be used to form a rectangle – the ones adjacent to the squares which have been cut out)

18

The number is a root of the equation, because

In this case the product of the roots is also

19

The fourth power of every natural number has 0, 1, or as its digit of ones In this case, when divided by 5, the remainder would be or

20 20

Atos did not shake hands with 70 people, Porthos did not shake hands with 60 people and Aramis did not shake hands with 50 people

In the worst case scenario, only Atos did not shake hands with those 70 people, Porthos did not shake hands with those 60 and Aramis did not shake hands with those 50 The total number is 180

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem The smallest natural number а, for which the equation || | | has exactly two solutions, is @ Find @

Problem 2. Find the number # of the natural numbers N, such that √ √

Problem 3. In ΔABC the side AB has been divided into equal parts Straight lines, parallel to AC, have been built through the points of division, which has created segments with their ends along the sides AB and BC of the triangle If the sum of these segments is # cm, calculate the length of side AC in centimeters Denote the answer using the symbol & Find &

Problem 4. The sum of the digits of the number equal to ⏟ is § Find §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ = 18

The equation || | | has a solution if If , the equation has exactly two solutions

Let We reach the conclusion that

| | or | | | | or | |

In order for there to be two solutions i.e If or the equation will have exactly two solutions The value of that we are looking for is the number 18

2 # =30 √ √ √ √

The number of natural numbers from to 35 is 30

3 & =15

Let us use to denote the segment that is closest to the vertex B It is a middle segment of a triangle where the next longest segment is , the one after is and the fourth is The sum of the four sections is 10 We have found that = cm In this case = = 15 cm

4 § = 27 ⏟

⏟

The sum we are looking for is 27

5 * =

Consider the following statement:

If among M coins only one is a fake (it is lighter) and M e a random number among the numbers and ,

then the minimum number of times we would need to weigh the coins in order to find the lighter coin is N

MATHEMATICS WITHOUT BORDERS 2015-2016

AUTUMN 2015: GROUP

Problem Which of the following is the largest number?

A) √ B) √ C) √ D) √

Problem Find the missing number in the equation

√

A) 25 B) 45 C) 90 D) 135

Problem The number of rational numbers in the sequence √ √ √ √ √ is:

A) 31 B) 16 C) 450 D) 225

Problem In the rectangular triangle , the hypotenuse √ and The area of the triangle is:

A) B) √ C) √ D) √

Problem If √ and √ , then the value of | | | | is:

А) B) C) √ D)- √

Problem A polygon has more than 100 diagonals The number of its sides is at least:

Problem When solving the same quadratic equation, three students got different roots: The first student got the numbers and as roots;

The second student - and 3; The third student - and

It turned out that each student got exactly one root of the equation right If the roots are и , then – is:

A) B) C) D)

Problem The numbers a and b are such that the expression has the smallest value possible The value is:

A) B) C) D)

Problem A rectangle is divided by two intersecting lines parallel to its sides Four smaller rectangles are formed, three of which have areas of and (see the diagram) Find the smallest possible value of the area of the fourth rectangle if

А) 𝑆 ×𝑆𝑆 B) 𝑆 ×𝑆𝑆 C) 𝑆 ×𝑆𝑆 D) other

Problem 10 How many are the natural numbers n, for which the number

is also a natural

number?

А) B) C) D)

Problem 11 If ( √ ) × (√ ) , what is the maximum value for √ ?

Problem 13 In trapezoid ) is the length of its leg AD and is the distance from the midpoint of AB to the leg AD (see the diagram) If АВ:DC=3:2, express the area of the trapezoid in terms of a and b

Problem 14 How many are the integers smaller than 2015, which can be represented as a sum of four consecutive integers?

Problem 15 What is the greatest number of cells 1x1 you can paint in a 11x11 square drawn on graph paper, so that no one box 2x2 did not have three of painted cells?

Problem 16 Find the integer , if √ √ √ √

Problem 17 The point D is of the median CM of the triangle ABC and it is such that СD=DM If the point E is an intersection of the straight line AD and the side BC, determine CE:CB

Problem 18 For which prime numbers and the roots of the equation are integers?

Problem 19 How many natural numbers are there, that are factors of ?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 D If we compare the numbers

√ √ √ √ we will find that the greatest of them is √ √ 2 D the number we are looking for is 135

3 B The numbers are √ √ √ √ √ They are as many as the odd numbers 1, 3, 5, , 29, 31 The odd numbers from to 31 inclusive, are 16

4 A Let us assume that С and С are respectively the height and the median of the side АВ In the rectangular С × √ The area of the triangle is equal to

×

5 A От √ <0 и √

| | | | | | | |

6 B The number of diagonals of a polygon with N angles can be determined by using the formula × By carrying out a check, we will find that

×

× Therefore the polygon has 16 angles

7 C The roots are either and 3, or and Therefore the value we are looking for is

8 B =

× The conclusion is a classic example:

9 B If we mark the areas of the rectangles with and , × ×

The smallest area would be 𝑆 ×𝑆

𝑆

10 C

=3+ is an integer for

The expression is a natural number only for and 11 1 ( √ ) × (√ ) √ or √

√ or √ √ {√ × √

12 3 The sum of the angles of a convex hexagon is 720◦ and each of them is smaller than 180◦ If of the angles are acute, then the sum of the other two would be greater than 360◦, i.e at least one of them would be greater than 180◦

13

is the midpoint of АВ The area of the triangle AMD is

DM is a

median in the triangle ABD Therefore the area of the triangle В D is

The area of the triangle АВD is

The areas of the triangles BCD and АВD are in the same ratio as AB and

CD Therefore the area of the triangle ABD would be The area of

the trapezium ABCD is

14 502 The sum of consecutive natural numbers is

when divided by leaves a remainder of

The numbers are 10, 14, 18, , 2010, 2014

(4 × × × × 502 in total

15 66 If we paint all odd columns, the number of colored squares would be

16 -2 First solution:

√ √ и √ √ |√ | |√ |

Second solution: If we raise to the second power we will get the

following result:

√ √ √ √ √ С или

√ √ √ √ <0

Third solution:

Apply the following formula:

√ √ √ √ √ √ 17 1:3

is the midpoint of АВ Therefore the areas of and are equal

If the areas of the triangles ACM and BCM are equal, then 2F=3S+F F=3S 4S=1:2

18 (2;2);

(2;3); (3;2)

If we apply Viet’s theorem, we will reach the conclusion that if a and b are the roots of the equation, then

( ) × ( )

If a>1 and b>1, then the solution is (2;2)

If a=1 or b=1 , then the solutions are (3;2) or (2;3)

19 16 × × , therefore the natural numbers which are divisors are 16

20

WINTER 2016: GROUP

Problem After we simplify the following expression and rewrite it in simple form, what would the coefficient of be?

×

A) B) C) D) other

Problem If √ × is a rational number, what is the smallest natural number N?

A) B) C) 14 D) 28

Problem How many different integers can be presented as “x” in the following inequality?

√

A) B) C) D) more than

Problem Equilateral triangle ABC is inscribed in a circle Point M is located on the smaller of the two arcs AC How many angles with vertices A, B, C and M are there on the diagram?

A) B) C) D)

Problem The differences of each two of three natural numbers are different What is the smallest possible sum of these three numbers?

Problem What is the remainder after dividing by 15?

A) B) C) 10 D) 15

Problem If and are the real roots of the equation , then | | is equal to:

A) B) √ C) √ D)

Problem The integers from to 50 have been written down on 50 cards What is the minimum number of cards that we would need to draw without looking in order to make sure we have drawn at least cards with prime numbers on them?

A) 38 B) 39 C) 40 D) 50

Problem An isosceles triangle has a perimeter of 56 cm, and two of its sides have a ratio of 3:2 What is the shortest possible side of the triangle?

A) 10 cm B) 14 cm C) 16 cm D) 24 cm

Problem 10 How many natural numbers that contain three different digits, one of which is a geometric mean of the other two, are there?

(Hint: If × , a is the geometric mean of b and c.)

A) B) 12 C) 18 D) 24

Problem 11 If the numbers a, b ∊ √ √ √ , how many ordered pairs of numbers (a,

b) are there, where either a + b, or a × b are natural numbers?

Problem 12 What is the ones digit of the number equal to the sum of the cubes of the numbers from to 101?

Problem 13 If ABCD is a square, and the point F is such that the triangle BCF is equilateral, what is the greatest possible measure that AFD can have?

Problem 14 How many proper irreducible fractions are there,whose numerator and denominator are natural numbers with a sum of 14?

Problem 16 The angles of the vertices B and C of the triangle ABC are respectively and The point M is interior to the triangle and Calculate

Problem 17 Calculate A if

×

Problem 18 Find the values of the parameter a, where the coordinate axes and the straight lines and form a trapezium

Problem 19 How many integers from 2000 to 2016 are there, that can not be values of the discriminant of the quadratic equation with integer coefficients?

Problem 20 On the diagram below you can see anisosceles triangle ABC with an area of and legs AC and BC, each with a length of 10 cm Point M is on the base AB, and points E and F are feet of altitudes from point M to the legs BC and AC Find the greatest possible value of the product ME×

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 А In the simple form of the polynomial

× ,

we can get the ninth degree from

× × × × ×

The coefficient is

2 C If √ × is a rational number, then the number equal to × × × must be a perfect square

Therefore the exponents must be even numbers and this is possible if N is at least 2×7

× × × × × × × × ×

3 B

√ ∊

4 C Just like the angles of the triangle ABC, the angles AMB and BMC are also equal to 60 degrees

5 B Let us assume that the numbers are 1, and n In this case the differences are The smallest value of n is Therefore the sum we are looking for is + + =

6 C × quotient + remainder

Hence the remainder is a natural number, that is a multiple of 5, but is smaller than 15 i.e it is either 0, or 10

The remainder is not 0, because cannot be divided by

The remainder is not 5, because in this case × × The number equal to × = ⏟

is not divisible by

Therefore the remainder is 10

7 B ×

The equation does not have real roots, because its discriminant <

Then и are real roots of the equation

8 А There are 15 primes among the numbers from to 50 Those numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 In this case we would need to take out at least 35+3=38 cards in order to make sure that there are at least primes among them

9 B There are two triangles that can satisfy the condition of the problem: A triangle with sides of 21, 21, 14; A triangle with sides of 16, 16, 24 10 D We are looking for the numbers containing the digits (1, 2, 4), (1, 3, 9), (2, 4,

8) and (4, 6, 9)

Their total number is 24: 124; 142; 214; 241; 412; 421; 139; 193; 391; 319; 913; 931; 248; 284; 482; 428; 824; 842 469; 496; 649; 694; 946; 964 11 4 The numbers are

( √ √ ) (√ √ ) ( √ √ ) ( √ √ )

12 1 The cubes of ten consecutive numbers have a ones digit equal to 1, 8, 7, 4, 5, 6, 3, 2, 9,

The ones digit of the sum of these cubes is

The sum of 100 consecutive numbers will end with the same digit as 10 × 5, i.e

The sum of the numbers from to 101 will end with the same digit as 10 × + 1, i.e

13 150 If point F is external to the square, then the angle we are trying to find would be 30 degrees, and if it is interior to the square, the angle would be 150 degrees

14 3 First way: We write down all proper fractions with 14 as a sum of their numerator and denominator and then we remove all reducible fractions

The fractions are 1/13, 3/11 and 5/9

Second way: The numerators of the fractions we are looking for would be all natural numbers, smaller than and non divisible by divisors of 14 other than and 14, i.e and The possible numerators would be 1, and

15 600 13 × × 17 × = 52 × 51, therefore the smallest natural number we are looking for is × × 50 = 600

13 × 17 × 600 = 50 × 51 ×

16 140 Point M is the center of the circle that encircles the triangle The angle we are looking for is central and

17 1,008

( × )

×

( × ) ×

18 0 and The straight lines and the coordinate axes would form a trapezium if the straight lines are parallel to one another In this case a =

A trapezium would also be formed when a = 0, in which case the straight line would be parallel to the x axis.

19 8 If the equation is , then If the number b is even, then D would be divisible by

If the number b is odd, then after dividing D by 4, there would be a remainder of

In this case the numbers among the integers from 2000 to 2016, that are not values of the discriminant, are: 2002, 2003, 2006, 2007, 2010, 2011, 2014, 2015

20 4 If x = MF and y = ME × × × × We are looking for the greatest value of the expression × ×

SPRING 2016: GROUP

Problem If × √ , then √

A) B) C) D)

Problem The numbers and are two of the four roots of the equation What is the sum of the other two roots?

A) B) C) D) other

Problem What is the value of the following expression? √ √ √

A) √ B) √ C) D)

Problem How many solutions does the following equation have? × √

A) B) C) D)

Problem There are points on the circumference of a circle What is the greatest possible number of right-angled triangles that have these points as their vertices?

A) 24 B) 30 C) 36 D)

Problem Two years ago A was twice older than B, and three years ago B was three times younger than A How old is A now?

A) 12 B) 10 C) D)

Problem For how many of the integers n can we claim that is divisible by ?

A) 0 B) C) D) more than

Problem What remainder is left when is divided by 21?

Problem The circle inscribed in the right-angled triangle ABC touches the hypotenuse AB at point

M If the radius of the circle is cm, and AM = cm, then AB – BC =

A) B) C) D)

Problem 10 In the graph | |, where is the parameter, and the coordinate axes determine a triangle with an area of What is the smallest possible value of the expression ?

A) B) C) D) other

Problem 11 If N and M are natural numbers, such that √ √ calculate N

Problem 12 The diagonals of a trapezium divide it into four triangles Three of the areas of the triangles are equal to respectively 4, and sq.cm What is the area of the trapezium?

Problem 13 What is the number of real roots of the equation | | ?

Problem 14 Six children, A, B, C, D, E and F, must stand in a row in such a way that A and B, C and D, E and F would always be standing next to one another In how many different ways can we this arrangement?

Problem 15 The polynomial is written in the following form: × × What is the value of ?

Problem 16 The numbers 201 and 235 leave the same remainder (14), when divided by Find Problem 17 If

√ √ √ √ , find A

Problem 18 The number of diagonals of a convex N-gon is 2015 What is the number N?

Hint:√

Problem 19 What would the last digit (the units digit) of the square of an integer be, if the digit before-last (the tens digit) is odd?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 C (√ ) (√ ) (√ ) √ √

2 C First way: We solve the equation to find that the roots are equal to 2, , and The sum we are looking for is

Second way: All equations of the kind that have and as solutions also have and – as solutions In this case the other two solutions are equal to and Their sum is equal to

3 D √ √ √ | √ | √ ( √ ) √

4 B × √ or √ the possible roots are 1, or The check shows that only is a root of the equation

5 А If we place the points two by two, so that they would be the edges of a diameter, we would get right-angled triangles with a common

hypotenuse for each diameter There are × right-angled triangles in total

6 D А B

3 years ago

2 years ago

The equation is

Now

7 B

and therefore

is an integer,

8 D

× ×

× ×

Тhe remainder after a division by 21 is

9 B If AB = c, BC = a, CA = b

10 B In the graph of | |, where is the parameter, and the coordinate axes determine the triangle with an area of

In this case the smallest possible value of the expression is

11 3 √ √ √

If √ is an irrational number Therefore We can now reach the conclusion that M =1 Therefore M+N =3

12 25 ABCD is the trapezium, O is the intersection of its diagonals, AB The areas of the triangles ADO and BCO are equal, therefore the possible areas of the four triangles are 4, 4, 6, 9; 4, 6, 6, 9; 4, 6, 9,

and 𝑆𝑆 and 𝑆𝑆 ,

therefore the areas of the triangles are 4, 6, and In this case the area of the trapezium is + + + = 25

13 2 | |

, if The roots are the numbers and

, if This equation has no real roots

14 48 We must arrange the X, Y and Z pairs, which are respectively made up of

A and B, C and D, E and F

15 13 The identity = × × is also true for Therefore

= × ×

16 17 We need a natural number greater than 14, which is a divisor of both and That number is 17

17 2

√ √ √ √

√ √

√ √ ( √ √ ) √ √

√ √

√

√ √

18 65 The number of diagonals is determined by the following formula:

19 6 , therefore the digit before last would be odd if the tens digit of is odd This would be possible if b or The ones digit would be

20 72 The lengths of the medians of the triangle ABC are, respectively,

cm, 12 cm and 15 cm The point M is the centroid of the triangle Let us duplicate the triangle and get the parallelogram АСВD, where the point N is the centroid of the triangle АВD

FINAL 2016: GROUP

Problem If √ √ √ , then √

A) B) C) D)

Problem A circle has been inscribed in the right-angled triangle ABC The circle touches the hypotenuse AB at the point M If AM = cm and BM = cm, then the area of the triangle is:

A) B) C) D)

Problem If the number a is rational and the number √ is also rational, then the value of b is:

A) B) C) D)

Problem What is the product of the real roots of the following equation?

A) B) C) D)

Problem Of all the triangles with sides a, b and c, such that , calculate the perimeter of the triangle with the largest area

A) 25 B) 24 C) 23 D) other answer

Problem Find the natural number , 97 < < 102, for which the expression is a perfect square of a natural number

A) 98 B) 99 C) 100 D) 101

Problem If the points M and N are the midpoints of the sides CD and DA of the parallelogram

ABCD, and the straight lines AM and BN intersect at the point P, then the =

Problem A square and a circle have a common part The area of the square, the area of the common part and area the circle relate to each other in a ratio of 4:1:17 What percentage of the figure's area is the area of the common part?

A) B) 10 C) 15 D) 20

Problem The number of rational numbers in the sequence √ √ √ √ √ is:

A) 44 B) 42 C) 22 D) 21

Problem 10 How many points (x, y) are there, with coordinates that are negative integers and ?

A) B) C) D) more than

Problem 11 The equation , where a and b are parameters, has a double root How many real roots does it have?

Problem 12 The acute-angled triangle ABC is inscribed in a circle with a center O and a radius of R If r is the radius of the circle tangent to the segments AO and BO, and the arc AB, and calculate

Problem 14 Present the following as an irreducible fraction ̅ ̅ ̅ ̅

Problem 15 The sides of the triangle ABC are: AB = cm, BC = cm and AC = cm Points K, M and N are the feet of the perpendiculars from point P to the sides AB, AC and BC respectively Calculate 3AK+4ВN+5СM

Hint: One of the classic theorems in geometry is that of the French mathematician Lazare Nicolas

Carnot: The perpendiculars drawn from points K, M and N to the sides AB, AC and BC of the triangle

ABC intersect at point P only when

Problem 16 If , and , calculate

Problem 17 In the numerical equation √ √ √ √ √ √ , known as “problem of the Indian mathematician Bhaskara” the last number has been replaced with the letter A Find

Problem 18 In a particular year three consecutive months had Sundays in them What are the possible sums of the days in these three consecutive months?

Problem 19 We are given a square with a side of 10 cm If we were to cut out smaller squares, each with a side of 1cm, from two of the opposite corners of the big square, what is the greatest possible number of rectangles with a size of cm by cm that we would be able to cut the newly formed figure into?

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 B √ √ √ √ √

2 A

If and are the lengths of the sides of the triangle and is its semiperimeter,

×

×

3 D The number √ would be rational if In this case

4 B

First we need to note that is not a root of the equation If ( )

×

It is no longer difficult to establish that the roots are and (-1) Their product is

5 B

Of all the triangles with sides a and b, the right-angled triangle with catheti and , and a hypotenuse equal to √ ∊ has the largest area

The parameter is equal to 24 cm

6 A

therefore if , then the expression would be a square of the natural number, i.e

We must note that if then is such that ;

We must also note that there is no perfect square between the squares of two consecutive natural numbers

7 A

Let us assume that the straight line intersects the continuation of the side

CD at the point Q

From the congruence of the triangles NDQ and NAB we get that

DQ = AB.

8 A

The area of the square, the common part, and the circle are respectively and The area of the whole figure is 20k The percentage we are looking for is

×

9 C

The numbers are

There are 22 numbers We must also note that 10 C The inequality is true for

11 2

The equation does not have any real roots has only real roots

12 30

and are tangent to the point D, аnd Е is tangent to and AO

In this case In the triangle the cathetus is twice as small as the hypotenuse We can conclude that

13 17

Therefore

In this case There are 17 integers of the interval

14

15 25 If

16

From the condition of the problem we get that и

17 √

√ √ √ √ √ √

√ √ √ √ √ √ √ √ ( √ ) √ √ √ =0 √

18 89 or 90

The possibilities are:

31+28+31=90 or 31+29+31=91; 28+31+30=89; 29 + 31 + 30=90 31+30+31=92;

30+31+30=91; 31+30+31=92; 30+31+31=92; 31+31+30=92; 31+30+31=92; 30+31+30=91; 31+30+31=92

Between 12 Sundays there are 12 + 11 × = 78 days; Therefore the following will remain of the sums

12 days or 13 days; 11 or 12; 14; 13; 14; 14; 14; 14; 13; 14 February is definitely among those months

19 48

Let us colour the board using a chessboard pattern The blocks which have been cut off are of the same color, and each of the rectangles 1х2 covers both colors

In this case it would be impossible to get 49 rectangles However it is possible to get 48

(Two of the squares can not be used to form a rectangle – the ones adjacent to the squares which have been cut out.)

20 2

TEAM COMPETITION – NESSEBAR, BULGARIA

MATHEMATICAL RELAY RACE

The answers to each problem are hidden behind the symbols @, #, &, § and * and are used in solving the following problem Each team, consisting of three students of the same age group, must solve the problems in 45 minutes and then fill a common answer sheet

GROUP

Problem The smallest integer that can take on the parameter a, and for which the equation ( √ ) × √ is only satisfied for one number, is @ Find @

Problem The height to the hypotenuse of a right-angled triangle is @ cm The smallest possible area of this triangle is # Find #

Problem After an awarding ceremony, the three winners were greeted by a total of # people The first was greeted by 80 people, the second was greeted by 60, and the third by 70 What is the smallest possible number of people that greeted all three winners? Denote the answer using the symbol & Find &

Problem The number of all non-negative integers k, for which the inequality √ has a solution presented as a positive integer, is § Find §

ANSWERS AND SHORT SOLUTIONS

Problem Answer Solution

1 @ = 10

The equation can have a maximum of roots, a and √

In order for there to be root, it would be enough for √ , or √ , i.e √

The value would be equal to 10

2 # = 100

The median of the hypotenuse is always at least 10 cm In this case the hypotenuse would be at least 20 cm

Therefore the smallest possible area of this triangle would be

3 & =10

The first was not greeted by 20 people, the second was not greeted by 40 and the third was not greeted by 30 In the worst case scenario:

Those who did not greeted the first winner did not greeted at least one of the other two either

Those who did greeted the second winner did not greeted at least one of the other two Those who did not greeted the third winner, did not

greeted at least one of the other two either In this case 20 + 40 + 30 = 90 people did not greeted the three winners The people who greeted all three would be at least 100 90 = 10

4 § =

The smallest possible k, for which the inequality √ has a solution, is and the greatest is

The number we are looking for is

4 * = 2/3

If A, B and C are the centers of the three circumferences, then AB = 2,

divisors (e h (Dirichlet's Principle)