The CENTRE for EDUCATION

in MATHEMATICS and COMPUTING

cemc.uwaterloo.ca

Pascal Contest
(Grade 9)
Tuesday, February 25, 2020
(in North America and South America)
Wednesday, February 26, 2020
(outside of North America and South America)

Time: 60 minutes ©2019 University of Waterloo

Calculating devices are allowed, provided that they do not have any of the following
features: (i) internet access, (ii) the ability to communicate with other devices,
(iii) information previously stored by students (such as formulas, programs, notes,
etc.), (iv) a computer algebra system, (v) dynamic geometry software.
Instructions
1. Do not open the Contest booklet until you are told to do so.
2. You may use rulers, compasses and paper for rough work.
3. Be sure that you understand the coding system for your response form. If you are not sure,
ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in
circles completely.
4. On your response form, print your school name and city/town in the box in the upper right
corner.
5. Be certain that you code your name, age, grade, and the Contest you are writing
in the response form. Only those who do so can be counted as eligible students.
6. This is a multiple-choice test. Each question is followed by five possible answers marked
A, B, C, D, and E. Only one of these is correct. After making your choice, fill in the
appropriate circle on the response form.
7. Scoring: Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
There is no penalty for an incorrect answer.
Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
8. Diagrams are not drawn to scale. They are intended as aids only.
9. When your supervisor tells you to begin, you will have sixty minutes of working time.
10. You may not write more than one of the Pascal, Cayley and Fermat Contests in any given
year.

Do not discuss the problems or solutions from this contest online for the next 48 hours.

The name, grade, school and location, and score range of some top-scoring students will be
published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location,
and score of some top-scoring students may be shared with other mathematical organizations
for other recognition opportunities.
 Scoring: There is no penalty for an incorrect answer.
Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth 5.

1. How many  symbols are in the figure?

(A) 24 (B) 20 (C) 15
(D) 17 (E) 25

2. The value of 0.8 + 0.02 is

(A) 0.28 (B) 8.02 (C) 0.82 (D) 0.16 (E) 0.01

3. If 2x + 6 = 16, the value of x + 4 is

(A) 7 (B) 8 (C) 9 (D) 15 (E) 13

4. When two positive integers are multiplied, the result is 24. When these two integers
are added, the result is 11. When the smaller integer is subtracted from the larger
integer, the result is
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

5. In the diagram, 4P QR has side lengths as shown. If

x = 10, the perimeter of 4P QR is P
(A) 29 (B) 31 (C) 25 x–1 7
(D) 27 (E) 23
Q x+1 R
24 − 2
6. The value of 3 is
2 −1
7 4
(A) 1 (B) 0 (C) 4 (D) 3 (E) 2

7. Ewan writes out a sequence where he counts by 11s starting at 3. The resulting
sequence is 3, 14, 25, 36, . . .. A number that will appear in Ewan’s sequence is
(A) 113 (B) 111 (C) 112 (D) 110 (E) 114

8. Matilda counted the birds that visited her bird feeder Birds at Matilda’s Bird Feeder
yesterday. She summarized the data in the bar graph 10
8
shown. The percentage of birds that were goldfinches is
Number

6
(A) 15% (B) 20% (C) 30% 4
2
(D) 45% (E) 60% 0
Goldfinch Sparrow Grackle
Type of Bird
9. In the diagram, three lines intersect at a point. What is
the value of x?
(A) 30 (B) 45 (C) 60 x°
x°
(D) 90 (E) 120
x°

10. Starting at 1:00 p.m., Jorge watched three movies. The first movie was 2 hours and
20 minutes long. He took a 20 minute break and then watched the second movie,
which was 1 hour and 45 minutes long. He again took a 20 minute break and then
watched the last movie, which was 2 hours and 10 minutes long. At what time did
the final movie end?
(A) 6:45 p.m. (B) 7:15 p.m. (C) 7:35 p.m. (D) 7:55 p.m. (E) 8:15 p.m.

Part B: Each correct answer is worth 6.

11. Anna thinks of an integer.

• It is not a multiple of three.

• It is not a perfect square.
• The sum of its digits is a prime number.

The integer that Anna is thinking of could be

(A) 12 (B) 14 (C) 16 (D) 21 (E) 26

12. Natalie and Harpreet are the same height. Jiayin’s height is 161 cm. The average
(mean) of the heights of Natalie, Harpreet and Jiayin is 171 cm. What is Natalie’s
height?
(A) 161 cm (B) 166 cm (C) 176 cm (D) 183 cm (E) 191 cm

13. The ratio of apples to bananas in a box is 3 : 2. The total number of apples and
bananas in the box cannot be equal to
(A) 40 (B) 175 (C) 55 (D) 160 (E) 72

14. A sequence of figures is formed using tiles. Each tile is an equilateral triangle with
side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed
by adding 1 tile to the previous figure. The first four figures are as shown:

How many tiles are used to form the figure in the sequence with perimeter 91 cm?
(A) 6 (B) 11 (C) 13 (D) 15 (E) 23
15. In the diagram, the large square has area 49, the medium
square has area 25, and the small square has area 9.
The region inside the small square is shaded. The region
between the large and medium squares is shaded. What
is the total area of the shaded regions?
(A) 33 (B) 58 (C) 45
(D) 25 (E) 13

16. Which of the following expressions is not equivalent to 3x + 6?

−9x − 18 1 2
(A) 3(x + 2) (B) (C) (3x) + (9)
−3 3 3
1
(D) (9x + 18) (E) 3x − 2(−3)
3
17. Ben participates in a prize draw. He receives one prize that is equally likely to be
worth $5, $10 or $20. Jamie participates in a different prize draw. She receives one
prize that is equally likely to be worth $30 or $40. What is the probability that the
total value of their prizes is exactly $50?
1 1 1 2 2
(A) 6 (B) 3 (C) 2 (D) 5 (E) 3

18. A positive integer n is a multiple of 7. The square root of n is between 17 and 18.
How many possible values of n are there?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

19. Each of the following 15 cards has a letter on one side and a positive integer on the
other side.

e 17 57 60 D

43 E 3 7 13

31 88 G H 21

What is the minimum number of cards that need to be turned over to check if the
following statement is true?

“If a card has a lower case letter on one side, then it has an odd integer on
the other side.”

(A) 11 (B) 9 (C) 7 (D) 5 (E) 3

20. A large 5 × 5 × 5 cube is formed using 125 small 1 × 1 × 1 cubes. There are three
central columns, each passing through the small cube at the very centre of the large
cube: one from top to bottom, one from front to back, and one from left to right.
All of the small cubes that make up these three columns are removed. What is the
surface area of the resulting solid?
(A) 204 (B) 206 (C) 200 (D) 196 (E) 192
Part C: Each correct answer is worth 8.

21. In the 4 × 5 grid shown, six of the 1 × 1 squares are not

intersected by either diagonal. When the two diagonals
of an 8×10 grid are drawn, how many of the 1×1 squares
are not intersected by either diagonal?
(A) 44 (B) 24 (C) 52
(D) 48 (E) 56

22. In the diagram, P Q is a diameter of a larger circle, point P

R is on P Q, and smaller semi-circles with diameters P R
and QR are drawn. If P R = 6 and QR = 4, what is the
ratio of the area of the shaded region to the area of the
unshaded region? R
(A) 4 : 9 (B) 2 : 3 (C) 3 : 5
(D) 2 : 5 (E) 1 : 2 Q

23. Ali, Bea, Che, and Deb compete in a checkers tournament. Each player plays each
other player exactly once. At the end of each game, either the two players tie or one
player wins and the other player loses. A player earns 5 points for a win, 0 points
for a loss, and 2 points for a tie. Exactly how many of the following final point
distributions are possible?

Player Points Player Points Player Points Player Points

Ali 15 Ali 10 Ali 15 Ali 12
Bea 7 Bea 10 Bea 5 Bea 10
Che 4 Che 4 Che 5 Che 5
Deb 2 Deb 4 Deb 2 Deb 0

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

24. Lucas chooses one, two or three different numbers from the list 2, 5, 7, 12, 19, 31, 50, 81
and writes down the sum of these numbers. (If Lucas chooses only one number, this
number is the sum.) How many different sums less than or equal to 100 are possible?
(A) 43 (B) 39 (C) 42 (D) 40 (E) 41

25. We call the pair (m, n) of positive integers a happy pair if the greatest common
divisor of m and n is a perfect square. For example, (20, 24) is a happy pair because
the greatest common divisor of 20 and 24 is 4. Suppose that k is a positive integer
such that (205 800, 35k) is a happy pair. The number of possible values of k with
k ≤ 2940 is
(A) 36 (B) 28 (C) 24 (D) 30 (E) 27
 (English)
Contest
Pascal
2020
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca

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