Đề thi Toán quốc tế CALGARY năm 2017

In the second race, Greg was given a 38 metre head start, and this time Greg won and finished 1 second ahead of Joey. Assuming both Greg and Joey ran at uniform speeds in both races, det[r]

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JUNIOR HIGH SCHOOL MATHEMATICS CONTEST

MAY 3, 2017

NAME: GENDER:

PLEASE PRINT (First name Last name)

SCHOOL: GRADE:

(9,8,7, )

• You have 90 minutes for the examination The test has two parts: PART A — short answer; and PART B — long answer The exam has pages including this one

• Each correct answer to PART A will score points You must put the answer in the space provided No part marks are given PART A has a total possible score of 45 points

• Each problem in PART B carries points You should show all your work Some credit for each problem is based on the clarity and completeness of your answer You should make it clear why the answer is correct PART B has a total possible score of 54 points

• You are permitted the use of rough paper Geome-try instruments are not necessary References includ-ing mathematical tables and formula sheets are not

permitted Simple calculators without programming or graphic capabilities are allowed Diagrams are not drawn to scale: they are intended as visual hints only

• When the teacher tells you to start work you should read all the problems and select those you have the best chance to first You should answer as many problems as possible, but you may not have time to answer all the problems

MARKERS’ USE ONLY

PART A ×5 B1 B2 B3 B4 B5 B6 TOTAL (max: 99)

BE SURE TO MARK YOUR NAME AND SCHOOL AT THE TOP OF THIS PAGE

PART A:

SHORT ANSWER QUESTIONS

(Place answers in the boxes provided)

A1

A1

If you place one die on a table, you can see five faces of it (the front, back, left, right and top) If you stack two dice on a table, then the number of visible faces is nine In a stack of three dice, the number of visible faces is thirteen, and so on How many dice you need to stack on a table (in a single stack) so that the number of visible faces is 101?

A2

A2

What is the perimeter (in cm) of the following figure?

A3

A3

The integer has the property that it is prime and one more than it (i.e., 6) is twice a prime (6 = 2×3) The next integer with this property is 13, since 13 is prime and one more than it (i.e., 14) is twice a prime (14 = 2×7) What is the next integer after 13 with this property?

A4

A5

A5

A number is multiplied by 32 and 32 is then added The result is divided by 32 and finally the original number is subtracted What is the answer?

A6

A6

In the game of pickleball, the winner scores points while the loser gets between and points (inclusive) Ruby plays games and gets a total of 50 points What is the smallest possible number of games she won?

A7

A7

Mary is 24 years old She is twice as old as Ann was when Mary was as old as Ann is now How old is Ann?

A8

A8

A belt runs tightly round three pulleys, each of diameter 40 cm The centre of the top pulley is 60 cm vertically above the centre of the second pulley, which is 80 cm horizontally from the centre of the rightmost one

What is the total length in cm of the belt?

A9

PART B:

LONG ANSWER QUESTIONS

B1

In the game Worm, Alice and Bob alternately connect pairs of adjacent dots on the shown grid with either a vertical line or a horizontal line Subsequent segments must start where the previous one ended and end at a dot not used before, forming aworm The player who cannot continue to build the worm (without it intersecting itself) loses

For example, if Alice’s first move is a1 – a2, Bob may then continue with either a2 – a3 or a2 – b2 Suppose Bob plays a2 – b2, and Alice then plays b2 – c2, followed by Bob playing c2 – c1 Then Alice will win with the move c1 – b1 since Bob has no remaining moves to continue building the worm

B2

We say that a by rectangle fitsnicelyinto a by square if the rectangle occupies exactly ten of the little squares in the by square

The diagram on the right shows the by square with two non-overlapping rectan-gles nicely placed in it

(a) How many by rectangles can you fit nicely into a by square without overlapping? The more rectangles you succeed in fitting into the square, the better your score will be

B3

(a) Write 2017 as a sum of two squares of positive integers