B5 The following hat is made with sides of the same length and with right angles: Starting with a square, we can create a new figure by replacing each side of the square by a hat, so tha[r]
(1)43rd ANNUAL CALGARY JUNIOR HIGH SCHOOL MATHEMATICS CONTEST
MAY 1st, 2019
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
(2)PART A: SHORT ANSWER QUESTIONS (Place answers in the boxes provided)
A1
6 A1 The perimeter of a rectangle with integer edge-lengths is 10cm What is the largest
area (in cm2) that the rectangle can have?
A2
50 A2 A store increases the price of a shirt by 10%, then reduces the cost by $10 The
price is then 90% of the original price Find the original price
A3
5pm A3 At 1PM Isaac had done 1/3 of his homework At 2PM he had done 1/2 his homework
He works at a constant rate all the time At what time did he finish his homework?
A4
11km A4 Melissa drives for 11 minutes, the first minute at 10 km/h, the second minute at 20
km/h, and so on until the 11th minute at 110 km/h What is the total distance (in km) she travelled?
A5
4 A5 Shaan takes a piece of paper 10cm square, and cuts circular holes of radius 1cm in
(3)A6
31 A6 Let A(0,0), B(3,5), C(3,0), D(5,0) and E(5,−5) be five points in the Cartesian
plane The pentagonABCDE and its reflection in thex-axis are combined to make a seven sided figure What is the area of this figure?
A7
9 A7 A lawn 10 metres square receives cm of rain over its entire surface Assuming that
the volume of each raindrop is cubic millimetre, the number of raindrops that fell on the lawn can be written as a one followed by a number of zeros How many zeros come after the 1?
A8
36 A8 The number 12 has the strange property that the next number (13) is prime, the
number after that (14) is twice a prime (since 14 = 2×7) and the number after that (15) is three times a prime (since 15 = 3×5) Find a number N bigger than 12 so thatN + is prime,N + is twice a prime, andN+ is three times a prime
A9
26 A9 Three positive integersa, b, c are such that 0< a < b < cand b−a, c−aand c−b
(4)PART B: LONG ANSWER QUESTIONS
B1 You and your friends want to order two 2-topping pizzas There is a selection of toppings, but one of your friends is picky and doesn’t want any toppings repeated, even if they are on different pizzas How many ways can you order the two pizzas, if each pizza has precisely two toppings?
Solution 1:
There are ways to choose the topping which is not selected for a pizza
Now we have toppings left, all of which must be used Start with any topping There are ways to pick which topping goes with this one The last pizza is deter-mined by this choice It follows that there are 5×3 =15 possible orders Solution 2:
We can list out every pizza explicitly Let the toppings be ABCDE
A B
C D A BC E A BD E A CD E B CD E A C
B D A C B E
A D B E
A D C E
B D C E A D
B C A EB C A ED C A ED C C DB E
There are15 possible orders Solution 3:
There are ways to choose the first topping for the first pizza, ways to choose the second topping for the first pizza, ways to choose the first topping for the second pizza, and ways to choose the second topping for the second pizza
However we must divide this by 2, as the order in which we made the pizzas didn’t matter, so the above counts double
Also, the order of the toppings on the first pizza didn’t matter, so the above counted another double Similarly, the order of the toppings on the second pizza didn’t matter, so the above counted again another double
(5)B2 An ant is walking along a spiral, as shown in the figure The spiral consists of eight quarter-circles joined together, so that:
• Arc AP1 has its centre in B
• Arc P1P2 has its centre in C
• Arc P2P3 has its centre in D
• Arc P3P4 has its centre in A
• Arc P4P5 has its centre in B
• Arc P5P6 has its centre in C
• Arc P6P7 has its centre in D
• Arc P7P8 has its centre in A
A B C
D P1
P5
P2
P6
P3 P7
P4
P8
Assume that the length of the side of squareABCD is one centimetre What is the total distance (in cm) travelled by the ant in walking along the spiral fromAtoP8?
Solution:
Each circular arc is a quarter of a circle, so it has length 2πr4 = πr2 , where r is the radius The radii of the arcs start at and increase by one until the largest arc at radius It follows that the total length is
= π·1 +
π·2 +· · ·
π·8 = π
2(1 + +· · ·+ 8) = π
2 8·9
2
= 18π
(6)B3 A Greek cross is a figure made up of five squares of side 1cm joined along the edges as pictured below:
A rectangular piece of flooring is tiled with 4×6 = 24 copies of a Greek cross, along with some fragments to fill up the edges as in the figure Find theexact length and width in cm of the rectangle
Solution:
By Pythagoras, the dashed line in
has length√5 It follows that the dimensions of the red rectangle in
are 6√5×4√5
Finally, from similar triangles, the altitude (the red line) in, has length
√
5 This is the last bit of the rectangle Therefore the dimensions are
(7)B4 Arrange the numbers to 15 in a row, so that each adjacent pair adds to a perfect square For example, you might try 15,1,3,6,10 which works so far because 15 + = 16 = 42, + = = 22, + = = 32, and + 10 = 16 = 42, but then you would get stuck because you can’t find a different number to add to 10 to give you a perfect square
Solution:
8 is adjacent only to and is adjacent only to 7, so that and are the ends of the row
• is adjacent to and 14, yielding 9, 7, 2, 14
• 14 is adjacent to and 11, yielding 9, 7, 2, 14, 11
• 11 is adjacent to and 14,
• is adjacent to and 11,
• is adjacent to and 12,
• 12 is adjacent to and 13,
• 13 is adjacent to and 12, yielding 9, 7, 2, 14, 11, 5, 4, 12, 13,
One can also proceed from the other end
• 15 is adjacent to and 10, yielding 8, 1, 15, 10
• 10 is adjacent to and 15, yielding 8, 1, 15, 10,
• is adjacent to and 10,
• is adjacent to two of 1, 6, and 13 is already surrounded by and 15, so that the chain is completed with 13, 3, 6, 10, 15, 1,
(8)B5 The following hat is made with sides of the same length and with right angles: Starting with a square, we can create a new figure by replacing each side of the square by a hat, so that each vertical side is replaced by a hat pointing outside the square and each horizontal side is replaced by a hat pointing inside the square, as shown below:
=⇒ =⇒
The following sequence of figures was created applying the same process to each new figure
=⇒ =⇒ =⇒
figure figure figure figure
Suppose the perimeter of the first figure, the square, is cm
(a) What is the perimeter (in cm) of figure 4? Solution:
Every figure has times as many sides as the previous one The new sides are one third of the previous This implies every new figure has perimeter equal to
5
3 of the previous one Therefore the perimeter of the fourth figure is
4cm×
3× ×
5 =
500
27 cm=18.5185 cm
(b) What is the area (in cm2) of figure 4? Solution:
(9)B6 Three large spheres sit on the floor of a gymnasium, touching in a row The centres of the spheres are pointsA, B, C in this order, with these points lying in a straight line The radii of spheres with centresAandBare metre and metres respectively
(a) Find the radius (in metres) of the sphere with centre C
(b) The spheres touch the floor at pointsX, Y, Z respectively Find distance XZ in metres
Solution:
The following diagram is to scale Extend AB to meet the ray from XY at a point Q
A
B
C
Z Y
X Q
SinceAX = and BY = 2, then by Midline TheoremAQ=AB= LetCZ =x Since trianglesQAX and QCZ are similar,AXQA = QCCZ, so
3 + + +x
x =
3 +x= 3x
x=
Therefore the radius of the sphere with centreC is 4m