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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]
(1)41st JUNIOR HIGH SCHOOL MATHEMATICS CONTEST MAY 3, 2018
NAME: GENDER:
PLEASE PRINT (First name Last name) (optional)
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
THE EXAM HAS PAGES INCLUDING THIS COVER PAGE
(2)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 13−41 = 121 If x1−x+11 = 1321 wherexis a positive integer, what isx?
A5
(3)A6
A6 A circle with centreOhas diameter AB A pointP 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 metres wide with streetlights on either side spaced metres apart If a dog runs from one streetlight to the next as shown, what is the total distance it runs in metres?
6m 6m 6m
6m 6m 6m
3m
4m
A8
A8 We knowx andyare positive integers such that 8x+ 9y= 127 Find the value ofx
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
1cm 2cm 2cm
(4)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
(5)(6)B3 The edge-length of the base of a square pyramid is cm The length of each of the four slanting edges is cm What is the pyramid’s height in cm?
8cm
(7)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◦
(8)(9)B6 Triangle ABC has edge-lengths BC = 13, AB = 14, CA = 15 The straight line
DE intersectsAC in the ratio 10 : and BC in the ratio :
Is DE parallel to AB, or 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 orBF
F?
F?
F?
F?
A B
C
E D
5
10
14