2015 Gauss Contest
Grade 7

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. In the diagram, the fraction of the circle that is shaded is equal to

   

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

2. The value of $10\times (5-2)$ is

   1. $13$
   2. $70$
   3. $7$
   4. $30$
   5. $50$

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

   

   Which runner ran the least distance?

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

4. The equal-arm scale shown is balanced.

   

   One has the same mass as

   1. 
   2. 
   3. 
   4. 
   5. 

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

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

6. The number of centimetres in 3.5 metres is

   1. $350$
   2. $30.5$
   3. $3.05$
   4. $3.50$
   5. $305$

7. The perimeter of the figure shown is

   

   1. $18$
   2. $17$
   3. $23$
   4. $20$
   5. $25$

8. Hannah scored 312 points during the basketball season. If her average (mean) was 13 points per game, how many games did she play?

   1. $24$
   2. $41$
   3. $17$
   4. $13$
   5. $30$

9. The number 6 has exactly four positive divisors: $1,2,3,$ and 6. How many positive divisors does 20 have?

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

10. How many different 3-digit whole numbers can be formed using the digits $4,7$ and $9$, assuming that no digit can be repeated in a number?

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

Part B: Each correct answer is worth 6.

11. At Gaussville School, a total of 480 students voted for their favourite subject. The results are summarized in the pie chart shown.

    

    How many students voted for math?

    1. $184$
    2. $192$
    3. $96$
    4. $144$
    5. $288$

12. A piece of paper is folded in half, creating two layers of paper. The paper is then folded in half again. This is continued until the paper has been folded in half a total of five times. The total number of layers of paper in the folded sheet is

    1. $16$
    2. $32$
    3. $25$
    4. $8$
    5. $64$

13. How many even whole numbers between 1 and 99 are multiples of 5?

    1. $5$
    2. $7$
    3. $9$
    4. $11$
    5. $13$

14. In the $3\times 3$ table shown, the numbers 1, 2 and 3 are placed so that each number occurs only once in each row and only once in each column.

    

    The value of $X+Y$ is

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

15. In the rectangle shown, the area of the shaded region is

    

    1. $60$cm${}^2$
    2. $20$cm${}^2$
    3. $30$cm${}^2$
    4. $40$cm${}^2$
    5. $50$cm${}^2$

16. You have exactly $4.40 (440 ¢) in quarters (25 ¢ coins), dimes (10 ¢ coins), and nickels (5 ¢ coins). You have the same number of each type of coin. How many dimes do you have?

    1. $20$
    2. $11$
    3. $10$
    4. $12$
    5. $4$

17. One corner of a cube is cut off, creating a new triangular face, as shown.

    

    How many edges does this new solid have?

    1. $18$
    2. $14$
    3. $24$
    4. $15$
    5. $13$

18. 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. $FG$

19. 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. ${126}^{th}$

20. In a triangle, the measure of one of the angles is ${45}^{\circ }$. The measures of the other two angles in the triangle are in the ratio $4:5$. What is the measure of the largest angle in the triangle?

    1. ${80}^{\circ }$
    2. ${90}^{\circ }$
    3. ${75}^{\circ }$
    4. ${85}^{\circ }$
    5. ${100}^{\circ }$

Part C: Each correct answer is worth 8.

21. The numbers 1 through 25 are arranged into 5 rows and 5 columns in the table below.

    1	2	3	4	5
    10	9	8	7	6
    11	12	13	14	15
    20	19	18	17	16
    21	22	23	24	25

    What is the largest possible sum that can be made using five of these numbers such that no two numbers come from the same row and no two numbers come from the same column?

    1. $75$
    2. $73$
    3. $71$
    4. $70$
    5. $68$

22. The width of a rectangle is doubled and the length is halved. This produces a square with a perimeter of $P$. What is the perimeter of the original rectangle?

    1. $P$
    2. $2P$
    3. $\frac{1}{2}P$
    4. $\frac{5}{4}P$
    5. $\frac{5}{2}P$

23. A palindrome is a positive integer that is the same when read forwards or backwards. The numbers 101 and 4554 are examples of palindromes. The ratio of the number of 4-digit palindromes to the number of 5-digit palindromes is

    1. $4:5$
    2. $5:2$
    3. $2:7$
    4. $4:3$
    5. $1:10$

24. In the diagram, rectangle $PQRS$ is made up of six identical squares. Points $U,V,W,X,Y,$ and $Z$ are midpoints of sides of the squares, as shown.

    

    Hide/Reveal Description of Points
    • Point U is on the side shared by the last two squares in the second row.
    • Point V is on the top side of the first square in the first row.
    • Point W is on the top side of the second square in the first row.
    • Point X is on the side shared by the last two squares in the first row.
    • Point Y is on the side shared by the squares in the third column.
    • Point Z is on the right side of the last square in the second row.

    ---

    Which of the following triangles has the greatest area?

    1. $PVU$
    2. $PXZ$
    3. $PVX$
    4. $PYS$
    5. $PQW$

25. Two different 2-digit positive integers are called a reversal pair if the position of the digits in the first integer is switched in the second integer. For example, 52 and 25 are a reversal pair. The integer 2015 has the property that it is equal to the product of three different prime numbers, two of which are a reversal pair. Including 2015, how many positive integers less than 10 000 have this same property?

    1. $18$
    2. $14$
    3. $20$
    4. $17$
    5. $19$