The CENTRE for EDUCATION

in MATHEMATICS and COMPUTING

cemc.uwaterloo.ca

Gauss Contest
Grade 7
(The Grade 8 Contest is on the reverse side)
Wednesday, May 17, 2023
(in North America and South America)
Thursday, May 18, 2023
(outside of North America and South America)

Time: 60 minutes ©2023 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.
You must not access any website or electronic materials other than this contest
system and the PDF of the contest paper.
Instructions
1. You may use rulers, compasses and paper for rough work.
2. After making your choice, fill in the appropriate answer in this online system. We recommend
that you also keep track of your answers on paper during the contest. If you need to change
your answer before submitting, you may change to a different answer or to “No answer” if
you would like to leave the question unanswered.
3. When you are finished answering questions, click “Submit” under the answers to Question 25
to submit your contest. Be sure to submit your responses within 60 minutes of when you
click the link to download the contest paper PDF.
4. 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.
5. Diagrams are not drawn to scale. They are intended as aids only.

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.
 Grade 7

Part A: Each correct answer is worth 5.

1. Kiyana gives half of her 24 grapes to a friend. How many grapes does she give away?
(A) 2 (B) 4 (C) 6 (D) 12 (E) 48

2. Based on the graph shown, which day of the week had

Daily Temperature
the highest temperature?

Temperature (˚C )
(A) Tuesday (B) Thursday (C) Friday 20

(D) Saturday (E) Sunday 10

0
Mon Tue Wed Thu Fri Sat Sun
Day of the Week

3. At a local farm, strawberries are sold at $16.50 for each basket. What is the cost to
buy 4 baskets of strawberries?
(A) $64.00 (B) $66.00 (C) $64.50 (D) $65.50 (E) $65.00

4. The temperature last night was −5◦ C. It is now 3◦ C. How many degrees warmer is
it now?
(A) 8◦ C (B) 3◦ C (C) 2◦ C (D) 13◦ C (E) 5◦ C

5. Sarah multiplied an integer by itself. Which of the following could be the result?
(A) 32 (B) 33 (C) 34 (D) 35 (E) 36

6. In the figure shown, P QRS has three sides of equal length

and SR = 16 cm. If the perimeter of P QRS is 40 cm, P Q
then the length of P Q is
(A) 6 cm (B) 7 cm (C) 8 cm S 16 cm R
(D) 9 cm (E) 10 cm

7. Which of the following is equal to a whole number?

52 52 52 52 52
(A) 5 (B) 7 (C) 4 (D) 3 (E) 6

8. A circle has a radius of 4 cm. A line segment joins two points on the circle. What is
the greatest possible length of the line segment?
(A) 10 cm (B) 8 cm (C) 4 cm (D) 12 cm (E) 6 cm

9. An integer is randomly chosen from the list 10, 11, 12, 13, 14, 15, 16, 17, 18, 19.
What is the probability that the chosen integer is even?
3 4 5 6 7
(A) 10 (B) 10 (C) 10 (D) 10 (E) 10

10. The grocery receipt shows the cost of three items before
tax is added. When a 5% tax is added to the cost of the Sponge $4.20
items, what is the total cost for the three items? Shampoo $7.60
(A) $15.16 (B) $15.08 (C) $15.22 Soap $3.20
(D) $15.75 (E) $15.38
 Grade 7

Part B: Each correct answer is worth 6.

11. In the diagram, BCD is a straight line segment. The

measure of ∠ABC is A
(A) 35◦ (B) 40◦ (C) 60◦ 35°
(D) 75◦ (E) 45◦
75°
D C B
12. Square W XY Z is divided into 100 small identical
squares. Some small squares are shaded and some are W Z
unshaded, as shown. How many more of the small squares
need to be shaded so that 75% of the area of W XY Z is
shaded?
(A) 3 (B) 4 (C) 5
(D) 6 (E) 7 X Y

13. In the diagram, the points (2, 1), (4, 1) and (2, 5) are three
y
vertices of a rectangle. What are the coordinates of the
fourth vertex of the rectangle? (2, 5)
(A) (5, 2) (B) (4, 4) (C) (1, 5)
(D) (4, 5) (E) (2, 4)
(2, 1) (4, 1)
x

14. The sum of two different prime numbers is 10. The product of these two numbers is
(A) 24 (B) 16 (C) 4 (D) 21 (E) 9

15. Suppose n is a number such that the mean (average) of the list of numbers 2, 9, 4, n, 2n
is equal to 6. What is the value of n?
(A) 9 (B) 12 (C) 10 (D) 5 (E) 6

16. Each number from 1 to 6 replaces one of the letters P, Q, R, S, T , and U . The sum of
P and Q is 5 and the difference between R and S is 5. If T is greater than U , what
number replaces the letter T ?
(A) 4 (B) 6 (C) 2 (D) 3 (E) 5

17. In the diagram, 4ABC is a right-angled isosceles

triangle. D is the midpoint of BC and E is the midpoint A
of AB. If AB = BC = 24 cm, what is the area of
4AED?
E
(A) 48 cm2 (B) 36 cm2 (C) 72 cm2
(D) 9 cm2 (E) 54 cm2
B D C
 Grade 7
18. A closed rectangular prism with height 8 cm is standing
on a face with dimensions 2 cm by 5 cm. The prism
contains water with a depth of 6 cm, as shown. When
the prism is tipped so that it stands on a face with the 6 cm
8 cm
greatest area, the depth of the water is
(A) 0.75 cm (B) 1 cm (C) 1.25 cm 5 cm
2 cm
(D) 1.5 cm (E) 1.75 cm

19. Two standard dice are rolled. The product of the two numbers rolled is calculated.
What is the probability that the ones digit of this product is 0?
11 2 1 1 5
(A) 36 (B) 9 (C) 36 (D) 6 (E) 36

a 2
20. How many pairs of positive integers a and b satisfy the equation + = 1?
7 b
(A) 4 (B) 1 (C) 0 (D) 5 (E) 2

Part C: Each correct answer is worth 8.

21. Eight-sided polygon ABCDEF GH has integer side

lengths. It can be divided into a rectangle and a square, A B
as shown. The area of the square is greater than the area G H
of the rectangle. The product of the two areas is equal
to 98. Which of the following could be the perimeter of F E
ABCDEF GH?
(A) 51 (B) 32 (C) 44
D C
(D) 34 (E) 33

22. A Gareth sequence is a sequence of numbers in which each number after the second
is the non-negative difference between the two previous numbers. For example, if a
Gareth sequence begins 15, 12, then

• the third number in the sequence is 15 − 12 = 3,

• the fourth number is 12 − 3 = 9,
• the fifth number is 9 − 3 = 6,

and so the resulting sequence is 15, 12, 3, 9, 6, . . . . If a Gareth sequence begins 10, 8,
what is the sum of the first 30 numbers in the sequence?
(A) 40 (B) 72 (C) 34 (D) 56 (E) 64

23. The digits from 1 to 9 are each used exactly once to write three one-digit integers
and three two-digit integers. The one-digit integers are equal to the length, width
and height of a rectangular prism. The two-digit integers are equal to the areas of
the faces of the same prism. What is the surface area of the rectangular prism?
(A) 176 (B) 184 (C) 186 (D) 198 (E) 212
 Grade 7
24. A circle is divided into six equal sections. Each section is
to be coloured with a single colour so that three sections red yellow
red green red blue
are red, one is blue, one is green, and one is yellow. Two
blue red green red
circles have the same colouring if one can be rotated to
yellow red
match the other. In the diagram, Figure 1 and Figure 2
have the same colouring, while Figure 1 and Figure 3 Figure 1 Figure 2

have different colourings. How many different colourings

red
are there for the circle? green red

(A) 14 (B) 12 (C) 24 red blue

yellow
(D) 10 (E) 20
Figure 3

25. A school trip offered its participants three activities: hiking, canoeing and swimming.
Attendance records show that of all participants

• 10 students participated in all three activities,

• 50% participated in at least hiking and canoeing,
• 60% participated in at least hiking and swimming,
• k% participated in at least canoeing and swimming, and
• no students participated in fewer than two activities.

If k is a positive integer, what is the sum of all possible values of k?

(A) 191 (B) 185 (C) 261 (D) 95 (E) 175