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Invitational World Youth Mathematics Intercity Competition 1999 Individual Contest Section A In this section, there are 12 questions Fill in the correct answer in the space provided at the end of each question Each correct answer is worth points Find the remainder when 122333444455555666666777777788888888999999999 is divided by Find the sum of the angles a, b, c and d in the following figure a c b d How many of the numbers 12 , 22 , , 1999 have odd numbers as their tens-digits? The height of a building is 60 metres At a certain moment during daytime, it casts a shadow of length 40 metres If a vertical pole of length metres is erected on the roof of the building, find the length of the shadow of the pole at the same moment Calculate 1999 − 19982 + 1997 − 19962 + ⋯ + 32 − 2 + 12 Among all four-digits numbers with as their thousands-digits, how many have exactly two identical digits? The diagram below shows an equilateral triangle of side The three circles touch each other and the sides of the triangle Find the radii of the circles Let a, b and c be positive integers The sum of 160 and the square of a is equal the sum of and the square of b The sum of 320 and the square of a is equal to the sum of and the square of b Find a Let x be a two-digit number Denote by f ( x) the sum of x and its digits minus the product of its digits Find the value of x which gives the largest possible value for f ( x) www.VNMATH.com 10 The diagram below shows a triangle ABC The perpendicular sides AB and AC have lengths 15 and respectively D and F are points on AB E and G are points on AC The segments CD, DE, EF and FG divide triangle ABC into five triangles of equal area The length of only one of these segments is integral What is that length? A G E C F D B 11 How many squares are formed by the grid lines in the diagram below? 12 There are two committees A and B Committee A had 13 members while committee B had members Each member is paid $6000 per day for attending the first 30 days of meetings, and $9000 per day thereafter Committee B met twice as many days as Committee A, and the expenditure on attendance were the same for the two committees If the total expenditure on attendance for these two committees was over $3000000, how much was it? Section B Answer the following questions, and show your detailed solution in the space provided after each question Each question is worth 20 points The diagram below shows a cubical wire framework of side An ant starts from a vertex and crawls along the sides of the framework If it does not repeat any part of its path and finally returns to the starting vertex, what is the longest possible length of the path it has travelled? A In the diagram below, BC is perpendicular to AC D is a point on BC such that BC = 4BD E is a point on AC such that AC = 8CE If AD = 164 and BE = 52, determine AB B D C E A When a particular six-digit number is multiplied by 2, 3, 4, and respectively, each of the products is still a six-digit number with the same digits as the original number but in a different order Find the original number www.VNMATH.com Invitational World Youth Mathematics Intercity Competition 1999 Team Contest (a) (b) Decompose 98 + + 54 + 32 + into prime factors Find two distinct prime factors of 230 + 320 The cards in a deck are numbered 1, 3, …, 2n − In the k-th step, ≤ k ≤ n, 2k − cards from the top of the deck are transferred to the bottom one at a time We want the new card on the top to be 2k − 1, which is then set aside After n steps, the whole deck should be set aside in increasing order How should the deck be stacked in order for this to happen, if (a) n=10; (b) n=30? (a) (b) Express as a sum of trhe reciprocals of distinct integers, one of which is Express as a sum of trhe reciprocals of distinct integers, one of which is 1999 (a) Show how to dissect a square into 1999 squares which may (b) have different sizes Dissect the first two shapes in the diagram below into the ten or fewer pieces which can be reassembled to form the third shape Figure (1) Figure (2) The diagram below shows a blank × table Each cell is to be filled in with one of the numbers 1, 2, 3, and 5, so there is exactly one number of each kind in each row, each column and each of the two long diagonals The score of a completed table is the sum of the numbers in the four shaded cells What is the highest possible score of a completed table?。 1999 IWYMIC Answers Individual Part I 540° 400 4 1999000 432 −1 13 90 10 10 11 190 12 14040000 Part II 16 109 142587 Team (a) 43165005 = × × 13 × 41 × 5399 (b) 13、61 (a) 11、1、5、7、15、3、13、19、9、19 13、1、47、33、25、3、57、45、49、55、43、5、19、39、11、17、21、 (b) 51、29、7、41、15、31、23、27、59、35、53、37、9 (a) 2、5、8、12、20、24 (b) 1×2、2×3、3×4、4×5、…、1998×1999、1999 (b) Figure A Figure B 17, for example 5 4 3 5 www.VNMATH.com Invitational World Youth Mathematics Intercity Competition 2000 Individual Contest Section A In this section, there are 12 questions Fill in the correct answer in the space provided at the end of each question Each correct answer is worth points Find the unit digit of 17 2000 The sum of four of the six fractions 1 1 1 , , , , and is equal to Find the 12 15 18 product of the other two fractions Find the smallest odd three-digit multiple of 11 whose hundreds digit is greater than its units digit Find the sum of all the integers between 150 and 650 such that when each is divided by 10, the remainder is Find the quotient when a four-thousand-digit number consisting of two thousand 1s followed by two thousand 2s is divided by a two-thousand-digit numbers every digit of which is 6 Find two unequal prime numbers p and q such that p+q=192 and 2p-q is as large as possible D is a point on the side BC of a triangle ABC such that AC=CD and ∠CAB = ∠ABC + 45° Find ∠BAD Let a, b, c, d and e be single-digit numbers If the square of the fifteen-digit number 100000035811ab1 is the twenty-nine-digit number 1000000cde2247482444265735361, find the value of a+b+c-d-e P is a point inside a rectangle ABCD If PA=4, PB=6 and PD=9, find PC 10 In the Celsius scale, water freezes at 0° and boils at 100° In the Sulesic scale, water freezes at 20° and boils at 160° Find the temperature in the Sulesic scale when it is 215° in the Celsius scale 11 The vertices of a square all lie on a circle Two adjacent vertices of another square lie on the same circle while the other two lie on one of its diameters Find the ratio of the area of the second square to the area of the first square 12 Ten positive integers are written in a row The sum of any three adjacent numbers is 20 The first number is and the ninth number is Find the fifth number Section B Answer the following questions, and show your detailed solution in the space provided after each question Each question is worth 20 points E is a point on the side AB and F is a point on the side CD of a square ABCD such that when the square is folded along EF, the new position A’ of A lies on BC Let D’ denote the new position of D and let G be the point of intersection of CF and A’D’ Prove that A’E+FG=A’G D F A D’ G E B A’ C Twenty distinct positive integers are written on the front and back of ten cards, one on each face of every card The sum of the two integers on each card is the same for all ten cards, and the sum of the ten integers on the front of the cards is equal to the sum of the ten integers on the back of the cards The integers on the front of nine of the cards are 2, 5, 17, 21, 24, 31, 35, 36 and 42 Find the integer on the front of the remaining card Given are two three-digit numbers a and b and a four-digit number c If the sums of the digits of the numbers a+b, b+c and c+a are all equal to 3, find the largest possible sum of the digits of the number a+b+c www.VNMATH.com Invitational World Youth Mathematics Intercity Competition 2000 Team Contest E is the midpoint of side BC of a square ABCD H is the point on AE such that BE = EH X is the point on AB such that AH = AX Prove that : AB × BX = AX Four non-negative integers have been entered in the following 5×5 table Fill in the remaining 21 spaces with positive integers so that the sum of all the numbers in each row and in each column is the same 82 79 103 For n ≥ , define an = 1000 + n Find the greatest value of the greatest common divisor of an and an +1 Five teachers predict the order of finish of five classes A, B, C, D and E in an examination Guesses First Second Third Fourth Fifth Teacher A B C D E Teacher E D A B C Teacher E B C D A Teacher C E D A B Teacher E B C A D After the examination, which produces no ties between classes, it turns out that each of two teachers guesses correctly the ranks of two of the classes but is wrong about the ranks of the other three The other three teachers are wrong about the rank of every class Find the order of finish of the classes     Find all triples (a, b, c) of positive integers such that a ≤ b ≤ c and 1 +  + 1 +  =  a  b  c  Each team is given 50 square cardboard pieces and 50 equilateral triangular cardboard pieces Using as many of these pieces as faces, construct a set of different convex polyhedra Two polyhedra with the same numbers of vertices, edges, square faces and triangular faces are not considered different 2000 IWYMIC Answers Individual Part I 1 166⋯ 667 1998 terms 101 180 231 19950 (181, 11) 22.5° 10 321 11 2:5 12 10 Part II 37 10800 Team 83 82 79 21 103 82 79 79 103 82 4001 (2, 4, 15)、(2, 5, 9)、(2, 6, 7)、(3, 3, 8) 及(3, 4, 5) 79 103 82 79 103 83 C、D、A、E、B Using all the pieces, we construct the following set of ten convex polyhedra Each is represented in two dimensions by what is known as its Schlegel diagram www.VNMATH.com Invitational World Youth Mathematics Intercity Competition 2001 Individual Contest Section A In this section, there are 10 questions Fill in the correct answer in the space provided at the end of each question Each correct answer is worth points Find all integers n such that + + + n is equal to a 3-digit number with identical digits In a convex pentagon ABCDE, ∠A = ∠B = 120°, EA = AB = BC = 2, and CD = DE = Find the area of the pentagon ABCDE D E C A B If I place a cm by cm square on a triangle, I can cover up to 60% of the triangle If I place the triangle on the square, I can cover up to of the square What is the area of the triangle? Find a set of four consecutive positive integers such that the smallest is a multiple of 5, the second is a multiple of 7, the third is a multiple of 9, and the largest is a multiple of 11 Between and o’clock, a lady looked at her watch She mistook the hour hand for the minute hand and vice versa As a result, she thought the time was approximately 55 minutes earlier Exactly how many minutes earlier was the mistaken time? In triangle ABC, the incircle touches the sides BC, CA and AB at D, E and F respectively If the radius of the incircle is units and if BD, CE and AF are consecutive integers, find the length of the three sides of ABC Determine all primes p for which there exists at least one pair of integers x and y such that p + = x and p + = y Find all real solutions of 3x − 18 x + 52 + x − 12 x + 162 = − x + x + 280 Simplify 12 − 24 + 39 − 104 − 12 + 24 + 39 + 104 into a single numerical value 10 Let M = 1010101…01 where the digit appears k times Find the least value of k so that 1001001001001 divides M ? 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest Individual Contest Time limit: 90 minutes Instructions: z Do not turn to the first page until you are told to so z Write down your name, your contestant number and your team's name on the answer sheet z Write down all answers on the answer sheet Only Arabic NUMERICAL answers are needed z Answer all 15 problems Each problem is worth 10 points and the total is 150 points For problems involving more than one answer, full credit will be given only if ALL answers are correct, no partial credit will be given There is no penalty for a wrong answer z Diagrams shown may not be drawn to scale z No calculator, calculating device or protractor is allowed z Answer the problems with pencil, blue or black ball pen z All papers shall be collected at the end of this test English Version The age of Max now times the age of Mini a year from now is the square of an integer The age of Max a year from now times the age of Mini now is also the square of an integer If Mini is years old now, and Max is now older than but younger than 100, how old is Max now? but less than of the children are boys What is the smallest possible number of children in this choir? In a choir, more than Each girl wants to ride a horse by herself, but there are only enough horses for 10 of them If the total number of legs of all the horses and girls is 990, how 13 many girls will have to wait for their turns? 23 57 = is incorrect However, if the same positive integer is 30 78 subtracted from each of 23, 30, 57 and 78, then it will be correct What is the number to be subtracted? Clearly, A team is to be chosen from girls and boys The only requirement is that it must contain at least girls How many different teams may be chosen? The product of five positive integers is 2014 How many different values are possible as their sum? A cat has caught three times as many black mice as white mice Each day, she eats black mice and white mice After a few days, there are 60 black mice and white mice left How many mice has the cat caught? M is the midpoint of the side CD of a square ABCD of side length 24 cm P is a point such that PA = PB = PM What is the minimum length, in cm, of PM? In a party, every two people shake hands except for Bob, who only shakes hands with some of the people No two people shake hands more than once If the total number of handshakes is 2014, with how many people does Bob shake hands? 10 The cost of a ticket for a concert is $26 for an adult, $18 for a youth and $10 for a child The total cost of a party of 131 people is $2014 How many more children than adults are in the party? 11 Two overlapping squares with parallel sides are such that the part common to the area of the larger square and both squares has an area of cm2 This is of the area of the smaller square What is the minimum perimeter, in cm, of the eight-sided figure formed by the overlapping squares? 12 The number of stars in the sky is × 12 + 98 × 102 + 998 × 1002 + ··· + 99···98 × 100···02 In the last term, there are 2014 copies of the digit in 99···98 and 2014 copies of the digit in 100···02 What is the sum of the digits of the number of stars? 13 In a triangle ABC, D is a point on BC and F is a point on AB The point K of reflection of B across DF is on the opposite side of AC to B AC intersects FK at P and DK at Q The total area of triangles AFP, PKQ and QDC is 10 cm2 If we of the area of ABC add to this the area of the quadrilateral DFPQ, we obtain What is the area, in cm2, of triangle ABC? A F P K Q B D C 14 After Nadia goes up a hill, she finds a level path on top of length 2.5 km At the end of it, she goes down the hill to a pond Later, she goes back along the same route Her walking speed is kph, but it decreases to kph going up the hill, and increases to kph going down the hill Her outward journey takes hour 36 minutes but her return journey takes hour 39 minutes She does not stop anywhere at any time What is the length, in km, from start point to the pond? 15 Five colours are available for the painting of the six faces of a cube One colour is used to paint two of the faces, while each of the other four colours is used to paint one face How many differently painted cubes can there be? Two cubes painting the same colours on corresponding faces after rotation or flip are not considered to be different 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM CONTEST TimeĈ60 minutes Instructions: z Do not turn to the first page until you are told to so z Remember to write down your team name in the space indicated on every page z There are 10 problems in the Team Contest, arranged in increasing order of difficulty Each question is printed on a separate sheet of paper Each problem is worth 40 points For Problems 1, 3, 5, and 9, only numerical answers are required Partial credits will not be given For Problems 2, 4, 6, and 10, full solutions are required Partial credits may be given z The four team members are allowed 10 minutes to discuss and distribute the first problems among themselves Each student must attempt at least one problem Each will then have 35 minutes to write the solutions of their allotted problem independently with no further discussion or exchange of problems The four team members are allowed 15 minutes to solve the last problems together z No calculator or calculating device or electronic devices are allowed z Answer must be in pencil or in blue or black ball point pen z All papers shall be collected at the end of this test English Version For Juries Use Only No Score Score 10 Total Sign by Jury 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM CONTEST 23rd July, 2014, Daejeon City, Korea TeamĈ ScoreĈ Exactly one pair of brackets is to be inserted into the expression 2×2−2×2−2×2−2×2−2×2 The left bracket must come before a and the right bracket after a Determine the largest possible value of the resulting expression Answer: 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM CONTEST rd 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ Divide the 18 numbers 1, 2, , 18 into nine pairs such that the sum of the two numbers in each pair is the square of an integer Answer: ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ The diagram below shows a right isosceles triangle partitioned into four shaded right isosceles triangles and a number of squares The side lengths of all the squares are positive integers The smallest ten squares are of side length cm Determine the total area, in cm2, of the four shaded triangles Answer: cm2 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ Each of the five families living in apartments 2, 3, 4, and 12 in a building will adopt one of five cats, whose ages are 1, 2, 3, and The adopting family’s apartment number must be divisible by the age of the cat Find all possible adoption schemes 1-year-old Answer: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) … 2-year-old 3-year-old 4-year-old 6-year-old 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ The positive integers 1, 2, , 2014 strung together form a very long multi-digit number 12345678910111213 201220132014 A seven-digit multiple of 11 is obtained by erasing all the digits before it and all the digits after it Determine the smallest possible value of this seven-digit number, which may not start with the digit Answer: 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ The diagram below shows a hexagonal configuration of 19 dots in an equilateral triangular grid (a) Determine the number of equilateral triangles of different sizes with all three vertices among the 19 dots Draw an equilateral triangle of each size on the diagram (b) Determine the number of equilateral triangles of each size Answer: (a) (b) different sizes 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ Each of teams A, B, C, D and E plays against every other team exactly once A win is worth points, a draw point and a loss points At the end of the tournament, no two teams have the same number of points A has the highest number of points despite losing to B Neither B nor C loses any game, but C has fewer points than D How many points does E have? Answer: points 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ P is a point inside a square ABCD of side length cm What is the largest possible value of the area, in cm2, of the smallest one among the six triangles PAB, PBC, PCD, PDA, PAC and PBD? Answer: cm2 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ A sequence of 2014 two-digit numbers is such that each is a multiple of 19 or 23, and the tens digit of any number starting from the second is equal to the units digit of the preceding number If the last number in the sequence is 23, determine the first number in the sequence Answer: 2014 Korea International Mathematics Competition 21~26 July, 2014, Daejeon City, Korea Elementary Mathematics International Contest TEAM rd CONTEST 23 July, 2014, Daejeon City, Korea TeamĈ ScoreĈ 10 There are ten real coins all of the same weight There is a fake coin which is heavier than a real coin, and another fake coin which is lighter than a real coin You cannot tell the coins apart Explain how, in four weighings using a balance, you may determine whether the total weight of the two fake coins is greater than, equal to or less than the total weight of two real coins? Answer: 2014 EMIC Answers Individual 49 45 704 144 15 61 10 43 11 30 12 2019 13 30 14 7.9 15 75 Team 36 1-year-old Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment Apartment (18, 7), (17, 8), (16, 9), (2, 14), (11, 5), (4, 12), (13, 3), (6, 10), (15, 1) 81 The possible adoption schemes are as follows 2-year-od 3-year-od 4-year-od 6-year-od Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment 12 Apartment Apartment Apartment Apartment 12 1001011 (a) 6, 95 (b) triangles whose side length is the largest For each triangle of the next size, such kind of triangles In a triangle of the next size, their number is 12 There are 12 triangles of the next size For each triangle of the next size, their number is 14 Finally, there are 24 triangles of the smallest size 8 10 Divide the coins into four groups A, B, C and D …… ... top to be 2k − 1, which is then set aside After n steps, the whole deck should be set aside in increasing order How should the deck be stacked in order for this to happen, if (a) n=10; (b) n=30?... get back from the Devil double the amount of what you pay If you lose, the Devil just keeps what you pay The Devil guarantees that you will only lose once, but the Devil decides which game you... is equal to a 3-digit number with identical digits In a convex pentagon ABCDE, ∠A = ∠B = 120°, EA = AB = BC = 2, and CD = DE = Find the area of the pentagon ABCDE D E C A B If I place a cm by cm