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[r]
Trang 14 2nd JUNIOR HIGH SCHOOL MATHEMATICS CONTEST
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
Trang 2PART A: SHORT ANSWER QUESTIONS (Place answers in
the boxes provided)
A1
18
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
11m2
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
8
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
11
A4 Notice that12−13 = 16 and 13−14 = 121 If x1−x+11 = 1321 where x is a positive integer,
what is x?
A5
5.844L
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.)
Trang 318◦
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
35m
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
55
A8 We know x and y are positive integers such that 8x + 9y = 127 Find the value of x
A9
8cm2
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
2cm
Trang 4PART 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
Solution:
1cm
1cm
If a corner circle is selected, there is exactly 4 choices for the second circle (distances 2cm,√5cm, 2√2cm and 1cm) indicated in blue If no corner circle is selected, there
is exactly 1 choice for the second circle (distance√2cm) indicated in red It follows there are 5 different distances
Trang 5B2 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? Solution:
Let a and b be the ages of Penny’s two brothers six years ago, where we may assume that a ≥ b Then, Penny’s age was ab, so her age now is ab + 6 But her brothers’ ages now must be a+6 and b+6, so Penny’s age now must be a+6+b+6 = a+b+12 Thus we know that,
ab + 6 = a + b + 12 , which simplifies to
ab − a − b = 6
By adding 1 to both sides and factoring, this equation becomes
(a − 1)(b − 1) = 7 The only integers solution to this equation that satisfy a ≥ b > 0 is when
a − 1 = 7 and b − 1 = 1 Thus the we have a = 8 and b = 2 These were the ages of the brothers six years ago, so the brothers are now aged 14 and 8 It follows that Penny’s age is now 22
Trang 6B3 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
?
Solution:
The length of the diagonal of the square base is 8√2 by Pythagoras’ Theorem Dividing this by 2, we can find the height by considering the triangle,
4√2cm
9cm
?
So by Pythagoras’ Theorem,
√
81 − 16 · 2 =√49 = 7 Therefore the height is 7cm
Trang 7B4 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◦
Solution:
2
5
17
9
10
14
12 7
The labelled points are distance apart 0.4m travelling clockwise around the table
So the ball will visit points 7, 12, 14, 5, 2, 17, 9, 10 before falling into the middle pocket of the bottom edge
Trang 8B5 Can 2018 be written as the difference of two square integers? If yes provide an example, if no explain why
Solution:
Suppose 2018 = x2− y2 for some positive integers x and y Then as 2018 factors as
2 · 1009,
(x − y)(x + y) = 2 · 1009 Because x − y and x + y have the same parity, then (x − y)(x + y) is either divisible
by 4 or is odd Either way this is impossible
Trang 9B6 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
5 4
10
14
9
Solution:
DG, CH, EI are perpendiculars from D, C, E, onto AB If CH = h, HA = x,
HB = y, then, By Pythagoras’ Theorem,
h2+ x2= 152, h2+ y2= 132, x2− y2 = 152− 132 = 56 and x + y = 14, so that x = HA = 9 and y = HB = 5
DG = 9
13h >
10
15h = EI
so that F is on the far left Triangles DGF , EIF are similar AI = 10159 = 6
HG = 1345 = 2013, so that AG = 9 +2013 = 13713
Let AF = z, so that GF = z + 137/13 and
z + 137/13
z + 6 =
F G
F I =
DG
EI =
9h/13 10h/15 =
27
26. 27z + 162 = 26z + 274 , so that AF = z = 112 Check by Menelaus:
10
5 ·
4
9 ·
14 + z
−z = −1
so 45z = 40(14 + z) It follows that z = 112 By either method AF =112
F
C
5 4