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]
Trang 143rd ANNUAL CALGARY JUNIOR HIGH SCHOOL
MATHEMATICS CONTEST
MAY 1st, 2019
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
Trang 2PART 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
it, not overlapping, so that more than half the area of the paper is removed What
is the smallest number of holes that he could cut?
Trang 331
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 pentagon ABCDE and its reflection in the x-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 1 cm of rain over its entire surface Assuming that
the volume of each raindrop is 1 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
that N + 1 is prime, N + 2 is twice a prime, and N + 3 is three times a prime
A9
26
A9 Three positive integers a, b, c are such that 0 < a < b < c and b − a, c − a and c − b
are all squares of integers What is the smallest possible value of c?
Trang 4PART B: LONG ANSWER QUESTIONS
B1 You and your friends want to order two 2-topping pizzas There is a selection of 5 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 5 ways to choose the topping which is not selected for a pizza
Now we have 4 toppings left, all of which must be used Start with any topping There are 3 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 are 15 possible orders
Solution 3:
There are 5 ways to choose the first topping for the first pizza, 4 ways to choose the second topping for the first pizza, 3 ways to choose the first topping for the second pizza, and 2 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
It follows that the total number of pizzas is 5×4×3×22×2×2 = 15 possible orders
Trang 5B2 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
D
P 1
P 5
P 2
P 6
P 4
P 8
Assume that the length of the side of square ABCD is one centimetre What is the total distance (in cm) travelled by the ant in walking along the spiral from A to P8? 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 1 and increase by one until the largest arc at radius 8 It follows that the total length is
= π · 1
π · 2
2 + · · ·
π · 8 2
= π
2(1 + 2 + · · · + 8)
= π 2
8 · 9 2
= 18π Therefore the total distance the ant travels is 18πcm
Trang 6B3 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 the exact 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, 1 2 has length
2
√
5 This is the last bit of the rectangle Therefore the dimensions are
Trang 7B4 Arrange the numbers 1 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 + 1 =
16 = 42, 1 + 3 = 4 = 22, 3 + 6 = 9 = 32, and 6 + 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 1 and 9 is adjacent only to 7, so that 8 and 9 are the ends of the row
• 2 is adjacent to 7 and 14, yielding 9, 7, 2, 14
• 14 is adjacent to 2 and 11, yielding 9, 7, 2, 14, 11
• 11 is adjacent to 5 and 14,
• 5 is adjacent to 4 and 11,
• 4 is adjacent to 5 and 12,
• 12 is adjacent to 4 and 13,
• 13 is adjacent to 3 and 12, yielding 9, 7, 2, 14, 11, 5, 4, 12, 13, 3
One can also proceed from the other end
• 15 is adjacent to 1 and 10, yielding 8, 1, 15, 10
• 10 is adjacent to 6 and 15, yielding 8, 1, 15, 10, 6
• 6 is adjacent to 3 and 10,
• 3 is adjacent to two of 1, 6, and 13 1 is already surrounded by 8 and 15, so that the chain is completed with 13, 3, 6, 10, 15, 1, 8
Therefore we obtain the chain 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8
Trang 8B5 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
Suppose the perimeter of the first figure, the square, is 4 cm
(a) What is the perimeter (in cm) of figure 4?
Solution:
Every figure has 5 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 × 5
3×
5
3 ×
5
3 =
500
27 cm = 18.5185 cm.
(b) What is the area (in cm2) of figure 4?
Solution:
In each step, the sides alternate adding and removing the same square of area from the figure Since the number of sides remains even (each step has 5 times more sides), this means the area remains constant It follows that the fourth figure has area 1cm2
Trang 9B6 Three large spheres sit on the floor of a gymnasium, touching in a row The centres
of the spheres are points A, B, C in this order, with these points lying in a straight line The radii of spheres with centres A and B are 1 metre and 2 metres respectively (a) Find the radius (in metres) of the sphere with centre C
(b) The spheres touch the floor at points X, 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
Since AX = 1 and BY = 2, then by Midline Theorem AQ = AB = 3
Let CZ = x Since triangles QAX and QCZ are similar,AXQA = QCCZ, so
3 + 3 + 2 + x
3 1
8 + x = 3x
x = 4 Therefore the radius of the sphere with centre C is 4m
By Pythagoras QX = 2√2 Since QC = 12, then by Pythagoras again we get
QZ = 8√2 It follows that XZ has length 6√2m