2024 Gauss Contest
Grade 7

Wednesday, May 15, 2024
(in North America and South America)

Thursday, May 16, 2024
(outside of North American and South America)

©2024 University of Waterloo

Instructions

Time: 1 hour

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.

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 answer sheet. If you are not sure, ask your teacher to explain it.
4. 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. When you have made your choice, enter the appropriate letter for that question on your answer sheet.
5. Scoring:
   1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
   2. There is no penalty for an incorrect answer.
   3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
6. Diagrams are not drawn to scale. They are intended as aids only.
7. When your supervisor instructs you to start, you will have sixty minutes of working time.

Part A: Each correct answer is worth 5.

1. When the four digits of $2024$ are added, the result is

   1. $10$
   2. $8$
   3. $224$
   4. $44$
   5. $16$

2. If $n=5$, the value of $n+2$ is

   1. $5$
   2. $6$
   3. $7$
   4. $8$
   5. $9$

3. Which of the following shapes has a vertical line of symmetry?

   1. 
   2. 
   3. 
   4. 
   5. 

4. Students at Gauss Middle School were asked to choose their favourite school day. The results are shown in the circle graph.

   

   Which day was chosen by exactly one-quarter of the students?

   1. Monday
   2. Tuesday
   3. Wednesday
   4. Thursday
   5. Friday

5. A square with side length $5$ has an area of

   1. $5$
   2. $10$
   3. $15$
   4. $16$
   5. $25$

6. In the diagram, $\mathrm{\angle }PQR$ is a straight angle.

   

   The value of $x$ is

   1. $54$
   2. $14$
   3. $24$
   4. $64$
   5. $34$

7. Katie completed two laps of a track without stopping. The first lap took $3$ minutes and $45$ seconds, and the second lap took $4$ minutes and $35$ seconds. What was her total time?

   1. $8$ minutes $30$ seconds
   2. $7$ minutes $50$ seconds
   3. $8$ minutes $50$ seconds
   4. $8$ minutes $20$ seconds
   5. $7$ minutes $40$ seconds

8. The sequence of the five symbols $◯$   $◂$   $⊠$   $△$   $★$ repeats to form the pattern:

   $◯$   $◂$   $⊠$   $△$   $★$   $◯$   $◂$   $⊠$   $△$   $★$   $\cdots$

   If the pattern is continued, the 23rd symbol in the pattern is

   1. $◯$
   2. $◂$
   3. $⊠$
   4. $△$
   5. $★$

9. Olivia cuts a $42$ cm length of string into $2$ cm pieces. Jeff cuts a $42$ cm length of string into $3$ cm pieces. How many more pieces of string does Olivia have than Jeff?

   1. $7$
   2. $8$
   3. $4$
   4. $6$
   5. $5$

10. A number is randomly chosen from the list $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. The probability that the chosen number is divisible by $2$, or by $3$, or by both $2$ and $3$, is

    1. ${\displaystyle \frac{4}{9}}$
    2. ${\displaystyle \frac{5}{9}}$
    3. ${\displaystyle \frac{6}{9}}$
    4. ${\displaystyle \frac{7}{9}}$
    5. ${\displaystyle \frac{8}{9}}$

Part B: Each correct answer is worth 6.

11. In the subtraction of the two-digit numbers shown, the letters $P$ and $Q$ each represent a single digit.

    

    The value of $P+Q$ is

    1. $7$
    2. $9$
    3. $11$
    4. $8$
    5. $10$

12. The length of a rectangle is twice its width. The perimeter of the rectangle is $120$ cm. The width of the rectangle is

    1. $20$ cm
    2. $60$ cm
    3. $30$ cm
    4. $50$ cm
    5. $10$ cm

13. Eloise purchased a number of water hand pumps to give to a charity. The mean (average) price was $\mathrm{\$}85$ per water pump. If Eloise spent a total of $\mathrm{\$}765$, how many water pumps did she purchase?

    1. $7$
    2. $8$
    3. $9$
    4. $10$
    5. $6$

14. The number $385$ has three prime factors. The sum of these prime factors is

    1. $21$
    2. $26$
    3. $25$
    4. $23$
    5. $22$

15. A circle has radius $2$. If the radius of the circle is tripled, the area of the original circle divided by the area of the new circle is

    1. ${\displaystyle \frac{1}{3}}$
    2. ${\displaystyle \frac{1}{6}}$
    3. ${\displaystyle \frac{1}{9}}$
    4. ${\displaystyle \frac{1}{2}}$
    5. ${\displaystyle \frac{1}{8}}$

16. Brett and Juanita each have a glass containing $300\text{ mL}$ of water. Brett pours half of his water out and then Juanita pours $20\mathrm{\%}$ of her water into Brett’s glass. What volume of water is now in Brett’s glass?

    1. $210\text{ mL}$
    2. $360\text{ mL}$
    3. $180\text{ mL}$
    4. $330\text{ mL}$
    5. $240\text{ mL}$

17. A circular spinner is divided into $12$ identical unshaded sections and $3$ identical shaded sections, as shown.

    

    Each unshaded section is $3$ times the size of each shaded section. An arrow is attached to the centre of the spinner. The arrow is spun once. What is the probability that the arrow stops in a shaded section?

    1. ${\displaystyle \frac{1}{15}}$
    2. ${\displaystyle \frac{1}{5}}$
    3. ${\displaystyle \frac{1}{12}}$
    4. ${\displaystyle \frac{1}{13}}$
    5. ${\displaystyle \frac{1}{4}}$

18. The Gaussbot factory assembles robots. Each robot comes in one of three colours: red, blue, or green. Each robot also has a number stamped on its head: $1$, $2$, $3$, or $4$. The $n$th robot assembled is the first robot to have the same colour and the same number as a previously assembled robot. What is the greatest possible value of $n$?

    1. $11$
    2. $12$
    3. $13$
    4. $7$
    5. $8$

19. Five different integers in a list have a median of $10$ and a range of $7$. What is the smallest possible integer in the list?

    1. $4$
    2. $5$
    3. $6$
    4. $7$
    5. $8$

20. A standing desk has $31$ height settings, numbered from the lowest height, $1$, to the highest height, $31$. Since the desk is not working properly, when the up button is pressed, the desk goes up $6$ settings at a time if possible, otherwise it does not move. When the down button is pressed, the desk goes down $4$ settings at a time if possible, otherwise it does not move. If the desk starts at setting number $1$, how many of the $31$ settings will the desk be able to stop at?

    1. $14$
    2. $16$
    3. $9$
    4. $15$
    5. $10$

Part C: Each correct answer is worth 8.

21. Five different integers are selected from $1$ to $6$ and one integer is placed into each of the five squares shown.

    

    The integers are placed so that the sum of the three integers in the vertical column is $7$, and the sum of the three integers in the horizontal row is $11$. Which integer does not appear in any square?

    1. $3$
    2. $4$
    3. $2$
    4. $6$
    5. $5$

22. In the diagram, $17$ toothpicks are used to make a $2$ by $3$ grid of squares.

    

    Of the toothpicks used, $10$ are outer toothpicks and $7$ are inner toothpicks. Suppose that toothpicks are used to make a $20$ by $24$ grid of squares. To the nearest percent, what percentage of toothpicks used are inner toothpicks?

    1. $88\mathrm{\%}$
    2. $95\mathrm{\%}$
    3. $93\mathrm{\%}$
    4. $70\mathrm{\%}$
    5. $91\mathrm{\%}$

23. A rectangular prism has integer edge lengths and has a volume of $V$. The six faces of the prism are painted and then the prism is cut into $1$ by $1$ by $1$ cubes. Of these cubes, $50$ cubes have no paint on them. What is the mean (average) of all possible values of $V$?

    1. $224$
    2. $310$
    3. $396$
    4. $288$
    5. $348$

24. A three-digit integer is an integer from $100$ to $999$, inclusive. A three-digit integer is called Tiny if no rearrangement of its digits gives a three-digit integer that is smaller. For example, $138$, $207$ and $566$ are Tiny, but $452$, $360$ and $727$ are not. How many three-digit integers are Tiny?

    1. $255$
    2. $201$
    3. $212$
    4. $234$
    5. $219$

25. Suppose that $$w,x,y,z,(x+y),(x+z),(234+z),\text{ and }(234-z)$$ are $8$ different prime numbers. If $w+x+y=234$, and each of $y$ and $z$ is less than $50$, the value of $w-y$ is

    1. $226$
    2. $150$
    3. $210$
    4. $174$
    5. $222$