Đề thi Toán quốc tế CALGARY năm 2018

B4 A rectangular pool table has dimensions 1.4m by 2.4m and six pockets, as shown (the center pockets on the horizontal sides are placed at the midpoint of their respective side). A ball[r]

41st JUNIOR HIGH SCHOOL MATHEMATICS CONTEST

MAY 3, 2018

PLEASE PRINT (First name Last name) (optional)

(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 9 pages including this one

• Each correct answer to PART A will score 5 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 9 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 do 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

THE EXAM HAS 9 PAGES INCLUDING THIS COVER PAGE

Please return the entire exam to your supervising teacher

PART A: SHORT ANSWER QUESTIONS (Place answers in

the boxes provided)

A1

A1 How many 4-digit numbers can be made by arranging the digits 2, 0, 1, and 8? (A 4-digit number cannot start with 0.)

A2

A2 A rectangle whose length and width are positive integers has perimeter 24 metres and its area (in square metres) is a prime number What is its area in square metres?

A3

A3 Two brothers and two sisters are waiting in line for ice cream How many ways can they line up so that the two brothers are not next to each other and the two sisters are not next to each other?

A4

A4 Notice that12−13 = 16 and 13−14 = 121 If x1−x+11 = 1321 where x is a positive integer, what is x?

A5

A5 Shaani brushes his teeth twice a day, every day Each time he brushes he uses 1 ml

of toothpaste How many litres of toothpaste will he have used from the morning of January 1st 2010 to the evening of December 31st 2018? (Keep in mind 2012 and

2016 were leap years, all other years were 365 days.)

A6 A circle with centre O has diameter AB A point P is placed on the circle such that

∠AOP = 36◦ (as in the diagram below) What is ∠OP B in degrees?

O

A

B

P

36◦

A7

A7 There is a path 4 metres wide with streetlights on either side spaced 6 metres apart

If a dog runs from one streetlight to the next as shown, what is the total distance it runs in metres?

3m

4m

A8

A8 We know x and y are positive integers such that 8x + 9y = 127 Find the value of x

A9

A9 The following diagram is a 5cm×4cm box containing several quarter circles of radius 2cm Find the area of the shaded region in square cm

2cm

PART B: LONG ANSWER QUESTIONS

B1 In the following diagram any two circles which are adjacent along the same side are 1cm apart, measured from their centres List all different distances (in cm) from the centre of one circle to the centre of another

1cm

1cm

B2 Penny’s age is the sum of the ages of her two brothers Six years ago, her age was the product of the ages of her two brothers How old is Penny and her brothers?

B3 The edge-length of the base of a square pyramid is 8 cm The length of each of the four slanting edges is 9 cm What is the pyramid’s height in cm?

8cm

8cm 9cm

?

B4 A rectangular pool table has dimensions 1.4m by 2.4m and six pockets, as shown (the center pockets on the horizontal sides are placed at the midpoint of their respective side) A ball is projected from the lower left corner at an angle of 45◦ with the sides of the table and bounces at the marked points shown in the figure, always at

45◦ The marked points are placed 0.4m around the table clockwise Will the ball continue indefinitely or will it fall into a pocket? If the latter is the case, which pocket?

45◦

45◦

45◦

45 ◦

45◦

B5 Can 2018 be written as the difference of two square integers? If yes provide an example, if no explain why

B6 Triangle ABC has edge-lengths BC = 13, AB = 14, CA = 15 The straight line

DE intersects AC in the ratio 10 : 5 and BC in the ratio 9 : 4

Is DE parallel to AB, or do the lines extending segments AB and DE intersect

AB in some point F , to the left or right? If DE does intersect AB, calculate the appropriate length, AF or BF

F ?

F ?

F ?

F ?

C

10

14

9