2015 Gauss Contest
Grade 8

Wednesday, May 13, 2015
(in North America and South America)

Thursday, May 14, 2015
(outside of North American and South America)

©2014 University of Waterloo

Instructions

Time: 60 minutes

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. The value of $1000+200-10+1$ is

   1. $1191$
   2. $1190$
   3. $1189$
   4. $1209$
   5. $1211$

2. What time is it 45 minutes after 10:20?

   1. 11:00
   2. 9:35
   3. 11:15
   4. 10:55
   5. 11:05

3. Which of the following is closest to 5 cm?

   1. The length of a full size school bus
   2. The height of a picnic table
   3. The height of an elephant
   4. The length of your foot
   5. The length of your thumb

4. The graph shows the total distance that each of five runners ran during a one-hour training session.

   

   Which runner ran the median distance?

   1. $\text{Phil}$
   2. $\text{Tom}$
   3. $\text{Pete}$
   4. $\text{Amal}$
   5. $\text{Sanjay}$

5. If $x+3=10$, what is the value of $5x+15$?

   1. $110$
   2. $35$
   3. $80$
   4. $27$
   5. $50$

6. A rectangle has a perimeter of 42 and a width of 4.

   

   What is its length?

   1. $19$
   2. $17$
   3. $34$
   4. $21$
   5. $38$

7. The equal-arm scale shown is balanced.

   

   One has the same mass as

   1. 
   2. 
   3. 
   4. 
   5. 

8. At the beginning of the summer, Aidan was 160 cm tall. At the end of the summer, he measured his height again and discovered that it had increased by $5\mathrm{\%}$. Measured in cm, what was his height at the end of summer?

   1. $168$
   2. $165$
   3. $160.8$
   4. $172$
   5. $170$

9. If $x=4$ and $y=2$, which of the following expressions gives the smallest value?

   1. $x+y$
   2. $xy$
   3. $x-y$
   4. $x÷y$
   5. $y÷x$

10. The number represented by $◻$ so that $\frac{1}{2}+\frac{1}{4}=\frac{◻}{12}$ is

    1. $3$
    2. $12$
    3. $9$
    4. $6$
    5. $15$

Part B: Each correct answer is worth 6.

11. In the diagram, the value of $x$ is

    

    1. $40$
    2. $50$
    3. $60$
    4. $70$
    5. $80$

12. Zara’s bicycle tire has a circumference of 1.5 m. If Zara travels 900 m on her bike, how many full rotations will her tire make?

    1. $900$
    2. $1350$
    3. $600$
    4. $450$
    5. $1200$

13. In the graph shown, which of the following represents the image of the line segment $PQ$ after a reflection across the $x$-axis?

    

    Hide/Reveal Description of Line Segments
    • The segment PQ is in the first quadrant. The point P is 3 units right and 3 units up from the origin. The point Q is 4 units right and 3 units up from the point P.
    • The segment PS is in the first quadrant. The point P is 3 units right and 3 units up from the origin. The point S is 4 units right and 3 units down from the point P.
    • The segment MN is in the second quadrant. The point M is 3 units left and 3 units up from the origin. The point N is 4 units left and 3 units up from the point M.
    • The segment FG is in the third quadrant. The point F is 3 units left and 3 units down from the origin. The point G is 4 units left and 3 units down from the point F.
    • The segment TU is in the fourth quadrant. The point T is 3 units right and 3 units down from the origin. The point U is 4 units right and 3 units down from the point T.
    • The segment WV is in the fourth quadrant. The point W is 3 units right and 6 units down from the origin. The point V is 4 units right and 3 units up from the point W.

    ---

    1. $PS$
    2. $TU$
    3. $MN$
    4. $WV$
    5. $FG$

14. Carolyn has a $5 bill, a $10 bill, a $20 bill, and a $50 bill in her wallet. She closes her eyes and removes one of the four bills from her wallet. What is the probability that the total value of the three bills left in her wallet is greater than $70?

    1. $0.5$
    2. $0.25$
    3. $0.75$
    4. $1$
    5. $0$

15. Two puppies, Walter and Stanley, are growing at different but constant rates. Walter’s mass is 12 kg and he is growing at a rate of 2 kg/month. Stanley’s mass is 6 kg and he is growing at a rate of 2.5 kg/month. What will Stanley’s mass be when it is equal to Walter’s?

    1. $24\text{kg}$
    2. $28\text{kg}$
    3. $32\text{kg}$
    4. $36\text{kg}$
    5. $42\text{kg}$

16. There is a square whose perimeter is the same as the perimeter of the triangle shown.

    

    The area of that square is
    1. $12.25{\text{cm}}^2$
    2. $196{\text{cm}}^2$
    3. $49{\text{cm}}^2$
    4. $36{\text{cm}}^2$
    5. $144{\text{cm}}^2$

17. When expressed as a repeating decimal, the fraction $\frac{1}{7}$ is written as $0.142857142857\dots$ (The 6 digits 142857 continue to repeat.) The digit in the third position to the right of the decimal point is 2. In which one of the following positions to the right of the decimal point will there also be a 2?

    1. ${119}^{th}$
    2. ${121}^{st}$
    3. ${123}^{rd}$
    4. ${125}^{th}$
    5. ${126}^{th}$

18. The operation $\mathrm{\Delta }$ is defined so that $a\mathrm{\Delta }b=a\times b+a+b$. For example, $2\mathrm{\Delta }5=2\times 5+2+5=17$. If $p\mathrm{\Delta }3=39$, the value of $p$ is

    1. $13$
    2. $12$
    3. $9$
    4. $10.5$
    5. $18$

19. There are 3 times as many boys as girls in a room. If 4 boys and 4 girls leave the room, then there will be 5 times as many boys as girls in the room. In total, how many boys and girls were in the room originally?

    1. $15$
    2. $20$
    3. $24$
    4. $32$
    5. $40$

20. A rectangle has side lengths 3 and 4. One of its vertices is at the point $(1,2)$. Which of the following could not be the coordinates of one of its other vertices?

    1. $(-3,-1)$
    2. $(1,-5)$
    3. $(5,-1)$
    4. $(-2,6)$
    5. $(1,-1)$

Part C: Each correct answer is worth 8.

21. In square $PQRS$, $M$ is the midpoint of $PS$ and $N$ is the midpoint of $SR$.

    

    If the area of $\mathrm{△}SMN$ is 18, then the area of $\mathrm{△}QMN$ is

    1. $36$
    2. $72$
    3. $90$
    4. $48$
    5. $54$

22. Exactly 120 tickets were sold for a concert. The tickets cost $12 each for adults, $10 each for seniors, and $6 each for children. The number of adult tickets sold was equal to the number of child tickets sold. Given that the total revenue from the ticket sales was $1100, the number of senior tickets sold was

    1. $110$
    2. $20$
    3. $40$
    4. $2$
    5. $18$

23. The list of integers $4,4,x,y,13$ has been arranged from least to greatest. How many different possible ordered pairs $(x,y)$ are there so that the average (mean) of these 5 integers is itself an integer?

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

24. Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, $t$ seconds, where $t$ is a positive integer. The product of the smallest and largest possible integer values of $t$ is

    1. $3705$
    2. $3762$
    3. $2816$
    4. $3640$
    5. $3696$

25. The alternating sum of the digits of 63 195 is $6-3+1-9+5=0$. In general, the alternating sum of the digits of a positive integer is found by taking its leftmost digit, subtracting the next digit to the right, adding the next digit to the right, then subtracting, and so on. A positive integer is divisible by 11 exactly when the alternating sum of its digits is divisible by 11. For example, $63195$ is divisible by 11 since the alternating sum of its digits is equal to 0, and 0 is divisible by 11. Similarly, $92807$ is divisible by 11 since the alternating sum of its digits is 22, but $60432$ is not divisible by 11 since the alternating sum of its digits is 9. Lynne forms a 7-digit integer by arranging the digits $1,2,3,4,5,6,7$ in random order. What is the probability that the integer is divisible by 11?

    1. $\frac{1}{35}$
    2. $\frac{5}{42}$
    3. $\frac{3}{35}$
    4. $\frac{1}{42}$
    5. $\frac{4}{35}$