Gauss Contest Grade 7 PDF Past Paper 2023

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: 1 hour ©2023 University of...

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: 1 hour ©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. 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the website, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. Kiyana gives half of her 24 grapes to a friend. How many grapes does she give away? (A) 2 2. (B) 4 (C) 6 (D) 12 (E) 48 Based on the graph shown, which day of the week had the highest temperature? (A) Tuesday (B) Thursday (C) Friday (D) Saturday (E) Sunday Temperature (˚C ) 1. Daily Temperature 20 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 4. 7. (B) 33 (A) 6 cm (B) 7 cm (D) 9 cm (E) 10 cm (E) $65.00 (C) 2◦ C (D) 13◦ C (E) 5◦ C (C) 34 (D) 35 (E) 36 P S (C) 8 cm Q 16 cm R Which of the following is equal to a whole number? 52 5 (B) 52 7 (C) 52 4 (D) 52 3 (E) 52 6 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 9. (B) 3◦ C 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, then the length of P Q is (A) 8. (D) $65.50 Sarah multiplied an integer by itself. Which of the following could be the result? (A) 32 6. (C) $64.50 The temperature last night was −5◦ C. It is now 3◦ C. How many degrees warmer is it now? (A) 8◦ C 5. (B) $66.00 (B) 8 cm (C) 4 cm (D) 12 cm (E) 6 cm 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? (A) 3 10 (B) 4 10 (C) 5 10 (D) 6 10 (E) 7 10 Grade 7 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 items, what is the total cost for the three items? (A) $15.16 (B) $15.08 (D) $15.75 (E) $15.38 Sponge $4.20 Shampoo $7.60 Soap $3.20 (C) $15.22 Part B: Each correct answer is worth 6. 11. In the diagram, BCD is a straight line segment. The measure of ∠ABC is (A) 35◦ (B) 40◦ (D) 75◦ (E) 45◦ A (C) 60◦ 35° 75° D 12. Square W XY Z is divided into 100 small identical squares. Some small squares are shaded and some are 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 (D) 6 (E) 7 (B) (4, 4) (D) (4, 5) (E) (2, 4) B W Z X Y (C) 5 13. In the diagram, the points (2, 1), (4, 1) and (2, 5) are three vertices of a rectangle. What are the coordinates of the fourth vertex of the rectangle? (A) (5, 2) C y (2, 5) (C) (1, 5) (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 of AB. If AB = BC = 24 cm, what is the area of 4AED? (A) 48 cm2 (B) 36 cm2 (D) 9 cm2 (E) 54 cm2 (C) 72 cm2 A E 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 greatest area, the depth of the water is (A) 0.75 cm (D) 1.5 cm (B) 1 cm 6 cm 8 cm (C) 1.25 cm 2 cm (E) 1.75 cm 5 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? (A) 11 36 (B) 2 9 (C) 1 36 (D) 1 6 (E) a 2 + = 1? 7 b (E) 2 20. How many pairs of positive integers a and b satisfy the equation (A) 4 (B) 1 (C) 0 5 36 (D) 5 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, as shown. The area of the square is greater than the area of the rectangle. The product of the two areas is equal to 98. Which of the following could be the perimeter of ABCDEF GH? (A) 51 (B) 32 (D) 34 (E) 33 (C) 44 A G H F E B C D 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 are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. In the diagram, Figure 1 and Figure 2 have the same colouring, while Figure 1 and Figure 3 have different colourings. How many different colourings are there for the circle? (A) 14 (D) 10 (B) 12 red blue (C) 24 yellow red green red red green blue red yellow red Figure 2 Figure 1 green red red red blue yellow (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 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 18, 2022 (in North America and South America) Thursday, May 19, 2022 (outside of North America and South America) Time: 1 hour ©2022 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the website, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. Which of the following numbers is closest to 10? (A) 1 2. (B) 5 (C) 8 (D) 13 The graph shows the number of hours that Gabe spent riding his bike from Monday to Friday. The day on which Gabe spent the greatest number of hours riding his bike is (A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday (E) 19 Number of Hours 1. 5 4 3 2 1 0 Time Spent Riding Bike Mon Tue Wed Thu Fri Day of the Week 3. If x is less than 5, a possible value of x could be (A) 7 4. 6. (C) 108 (D) 12 (E) 23 In a sequence of numbers, the first term is 3. Each new term is obtained by adding 5 to the previous term. The first four terms are 3, 8, 13, 18. What are the next three terms in the sequence? (A) 25, 30, 35 5. (B) 0 (B) 5, 10, 15 (C) 23, 28, 33 (D) 23, 33, 43 (E) 19, 20, 21 The faces of a cube are labelled with 1, 2, 3, 4, 5, and 6 dots. Three of the faces are shown. What is the total number of dots on the other three faces? (A) 6 (B) 8 (D) 12 (E) 15 (C) 10 In the diagram, ∠ABC = 90◦. The value of x is (A) 46 (B) 22 (D) 42 (E) 54 A (C) 36 44° B x° C Grade 7 7. The singers in Saura’s choir have heights 148 cm, 141 cm, 172.5 cm, 168 cm, 151.5 cm, 183.5 cm, and 178.5 cm What is the range of their heights? (A) 42.5 cm 8. (B) 27 cm (C) 52.5 cm (D) 37.5 cm (E) 31.5 cm In the diagram, the point (3, −4) is labelled (A) P (B) Q (D) S (E) T y (C) R 5 4 R P 3 2 1 –5 –4 –3 –2 –1 –1 –2 Q 9. 1 2 –3 –4 3 4 T –5 5 x S While using a skipping rope, Emily jumps 52 times in 60 seconds. Jumping at this same rate, how many times does Emily jump in 75 seconds? (A) 66 (B) 52 (C) 65 (D) 67 (E) 73 10. A dime is worth $0.10 and a quarter is worth $0.25. Terry has a jar that contains $1.00 worth of dimes and $1.00 worth of quarters. If he randomly removes one coin from the jar, what is the probability that it is a dime? (A) 1 10 (B) 2 7 (C) 10 11 (D) 2 5 (E) 5 7 Part B: Each correct answer is worth 6. 11. The sum of the prime factors of 42 is (A) 23 (B) 43 (C) 12 (D) 17 12. In the diagram, 4P QR is isosceles with P Q = P R, and QRST is a rectangle. If ∠QP R = 70◦ , ∠P QR = x◦ , and ∠RQT = y ◦ , the value of x + y is (A) 70 (B) 90 (D) 145 (E) 60 (C) 160 (E) 13 P 70˚ x˚ Q y ˚ R T S 13. How many two-digit numbers have at least one digit that is a 4? (A) 17 (B) 11 (C) 18 (D) 10 Y (E) 19 Grade 7 14. Three identical squares form rectangle W XY Z, as shown. The perimeter of W XY Z is 56 m. The area of W XY Z is (A) 66 m2 (B) 147 m2 (D) 196 m2 (E) 348 m2 (C) 168 m2 W X Z P Y 15. A public holiday is always celebrated on the third Wednesday of a certain month. In that month, the holiday cannot occur on which of the following days? (A) 16th (B) 22nd (C) 18th (D) 19th (E) 21st 16. A standard fair coin is tossed three times. What is the probability that the three outcomes are all the same? (A) 1 2 (B) 3 16 (C) 1 4 (D) 5 16 (E) 17. In the sum shown, each letter represents a digit from 1 to 9, inclusive. The value of P + Q + R is (A) 13 (B) 14 (D) 16 (E) 17 (C) 15 + 1 8 QR PPP PPP 2 0 2 2 18. Box A contains one 100 g block, one 20 g block and three 5 g blocks. Box B contains one 50 g block and three 10 g blocks. Jasmine moves some of the blocks from Box A to Box B and some of the blocks from Box B to Box A. After these moves, Box A contains 65 g less than it originally did and Box B contains 65 g more. What is the fewest number of blocks that Jasmine could have moved from Box A to Box B? (A) 3 (B) 4 (C) 2 (D) 5 (E) 1 19. In a candy dish, the ratio of red to blue candies is 3 : 5. When 3 blue candies are removed, the ratio of red to blue candies becomes 2 : 3. How many more blue candies than red candies were in the dish before any candies were removed? (A) 10 (B) 12 (C) 6 (D) 8 (E) 14 20. Four friends, standing in a row for a picture, are in the following order: Anyu Brad Chi Diego The friends then move so that Anyu is not in the 1st position, Brad is not in the 2nd position, Chi is not in the 3rd position, and Diego is not in the 4th position. In how many ways can the friends do this? That is, in how many ways can they rearrange themselves so that each person is not in their original position? (A) 6 (B) 9 (C) 12 (D) 18 (E) 24 Grade 7 Part C: Each correct answer is worth 8. A B D C 21. Square ABCD is divided into four identical smaller squares, which are further divided into triangles, as shown. What fraction of ABCD is shaded? (A) (D) 1 4 3 8 (B) (E) 15 32 7 8 (C) 7 16 22. In the list p, q, r, s, t, u, v, w, each letter represents a positive integer. The sum of the values of each group of four consecutive letters in the list is 35. If q + v = 14, the largest possible value of p is (A) 15 (B) 19 (C) 20 (D) 23 (E) 26 23. Ishari places the letters A, B, C, D, E, F clockwise around a circle, starting with A at the top. Jaxon writes a list beginning with A, and then, moving clockwise around the circle, writes down every third letter that he has not yet written. Doing this, Jaxon’s list is A, D, B, F , C, E. Katharina mixes up the letters L, M , N , O, P , Q, R, S and places them in the mixed-up order around a circle, starting with L at the top. Jaxon writes a list beginning with L and then again moving clockwise around the circle, writes down every third letter that he has not yet written. Jaxon’s list is L, M , N , O, P , Q, R, S. Starting with L, what was Katharina’s clockwise order? (A) L, O, R, N, S, Q, M, P (B) L, Q, O, M, S, R, N, P (D) L, M, N, O, P, Q, R, S (E) L, O, R, M, Q, P, N, S (C) L, R, O, M, S, Q, N, P 24. A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 32 523 is a palindrome. How many palindromes greater than 10 000 and less than 100 000 are multiples of 18? (A) 41 (B) 42 (C) 43 (D) 44 (E) 45 25. Arjun has a bag that contains 5 balls and Becca has a bag that contains 3 balls. Arjun’s bag contains 2 red balls, 1 green, 1 yellow, and 1 violet ball. Becca’s bag contains 2 black balls and 1 orange ball. Arjun randomly chooses 1 ball from his bag and puts it into Becca’s bag. Becca then randomly chooses 1 ball from her bag and puts it into Arjun’s bag. Again, Arjun randomly chooses 1 ball from his bag and puts it into Becca’s bag. After these exchanges, there are 4 balls in each bag. What is the probability that each bag contains exactly 3 different colours of balls? (A) 3 10 (B) 6 25 (C) 9 50 (D) 3 25 (E) 9 25 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 12, 2021 (in North America and South America) Thursday, May 13, 2021 (outside of North America and South America) Time: 1 hour ©2021 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the website, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. When the five numbers 10 000, 1, 10, 100, and 1000 are arranged from largest to smallest, the middle number is (A) 10 000 (B) 1 (C) 10 (D) 100 (E) 1000 2. What is the perimeter of the square shown? (A) 20 cm (B) 8 cm (D) 50 cm (E) 15 cm (C) 5 cm 5 cm What value goes in the box to make the equation 5 +  = 10 + 20 true? (A) 30 (B) 15 (C) 35 (D) 20 (E) 25 4. The number of hours spent by five students on homework is shown on the graph. Which two students, adding their individual times together, spent the same amount of time on homework as Dan? (A) Joe and Grace (B) Joe and Bob (C) Bob and Grace (D) Dan and Bob (E) Susie and Grace 5. Which of the following fractions is closest to 0? (A) 12 (B) 81 (C) 13 (D) Time Spent on Homework 6 5 Grace Bob Joe 0 Susie 4 3 2 1 Dan Number of Hours 3. Student 1 6 (E) 1 9 6. A bag contains a number of candies. The probability of Judith choosing a red candy from this bag is 65. The total number of candies in the bag could be (A) 3 (B) 10 (C) 17 (D) 6 (E) 7 7. In the graph shown, which of the following statements is true about the coordinates of the point P (x, y)? y (A) The values of both x and y are positive. (B) The value of x is positive and the value of y is negative. (C) The value of x is negative and the value of y is positive. (D) The values of both x and y are negative. (E) The value of x is 0 and the value of y is negative. The line graph shows the distance that Andrew walked over time. How long did it take Andrew to walk the first 2 km? (A) 15 minutes (B) 1 hour, 15 minutes (C) 1 hour, 45 minutes (D) 2 hours (E) 45 minutes P (x, y) Andrew’s Walking Distance Distance (km) 8. x 3 2 1 1 2 3 Time (hours) Grade 7 9. A list of five numbers repeats to form the pattern 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9,... What is the 221st number in the pattern? (A) 5 (B) 6 (C) 7 (D) 8 10. An ant begins its path at A, travels only right or down, and remains on the line segments shown. The number of different paths from A to C that pass through B is (A) 2 (B) 3 (D) 5 (E) 6 (E) 9 A B (C) 4 C Part B: Each correct answer is worth 6. 11. Laila writes a list of numbers. Her first number is 4. Each number after the first is 7 more than the previous number. Which of the following numbers appears in Laila’s list? (A) 45 (B) 46 (C) 47 (D) 48 (E) 49 12. The letter A has a vertical line of symmetry and the letter B does not. How many of the letters H L O R X D P E have a vertical line of symmetry? (A) 1 (B) 2 (C) 3 (D) 4 13. In the diagram, AB and CD intersect at E. If 4BCE is equilateral and 4ADE is a right-angled triangle, what is the value of x? (A) 90 (D) 45 (B) 60 (E) 5 B D xº E (C) 25 C A (E) 30 14. Which of the following is the sum of three consecutive integers? (A) 17 (B) 11 (C) 25 (D) 21 (E) 8 15. A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. What is the sum of the digits of the next palindrome greater than 13931? (A) 14 (B) 11 (C) 19 (D) 10 (E) 8 16. The number 6 has exactly 4 positive factors and the number 9 has exactly 3 positive factors. How many numbers in the list 14, 21, 28, 35, 42 have exactly 4 positive factors? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 17. The original price of a shirt is reduced by 50% to obtain a second price. The store advertises an additional sale, and so this second price is reduced by 40% to obtain a third price. What is the discount of the third price off the original price? (A) 80% (B) 10% (C) 70% (D) 65% 18. In the diagram, 4ABC is isosceles. M is on BC so that BM = M C. If the perimeter of 4ABC is 64 and the perimeter of 4ABM is 40, what is the length of AM ? (A) 10 (B) 8 (D) 12 (E) 24 (C) 16 B (E) 45% A M C Grade 7 19. Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive? (A) 36 (B) 32 (D) 34 (E) 38 5 5 (C) 30 20. Two standard dice are rolled. What is the probability that the sum of the numbers on the top faces is a prime number? 5 7 (A) 12 (B) 12 (C) 12 (D) 56 (E) 13 Part C: Each correct answer is worth 8. 21. A large number is written with a one followed by many zeros (1000... 000). When 1 is subtracted from this number, the sum of the digits in the result is 252. How many zeros are in the original number? (A) 27 (B) 28 (C) 29 (D) 42 (E) 252 22. In the diagram shown, each figure after Figure 1 is formed by joining two rectangles to the bottom of the previous figure. Each individual rectangle has dimensions 10 cm by 5 cm. If Figure n has a perimeter of 710 cm, the value of n is (A) 29 (B) 43 (D) 172 (E) 65 10 cm Figure 1 5 cm Figure 2 (C) 66 Figure 3 23. To encode a message, James first replaces each letter within the message with its corresponding number, where A = 1, B = 2, · · · , Y = 25, and Z = 26. Next, James multiplies each number by 3 and then subtracts 5, and continues this process a total of n times. For example, when n = 2 the letter D is encoded to the number 16. If James encoded a four letter message to the four numbers 367 205 853 1339, what is the value of n that he used? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 24. How many different pairs of positive whole numbers have a greatest common factor of 4 and a lowest common multiple of 4620? (A) 4 (B) 5 (C) 7 (D) 8 (E) 11 25. Jonas has 1728 copies of a 1 × 1 × 1 cube with the net shown, where c is a positive integer and c < 100. Using these 1728 cubes, Jonas builds a large 12×12×12 cube in such a way that the sum of the numbers on the exterior faces is as large as possible. For some values of c, the sum of the numbers on the exterior faces is between 80 000 and 85 000. The number of such values of c is (A) 39 (B) 38 (D) 36 (E) 35 (C) 37 c c c 100 c c 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 13, 2020 (in North America and South America) Thursday, May 14, 2020 (outside of North America and South America) Time: 1 hour ©2020 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. One pen costs $2. What is the cost of 10 pens? (A) $4 2. 3. (A) (4, 0) (B) (2, 2) (D) (4, 4) (E) (2, 4) 7. 9. 5 4 P 3 2 (C) (2, 0) 1 1 2 3 4 5 x (B) 100 (C) 100 000 (D) 1000 (E) 10 (B) 3 (C) 5 (D) 2 (E) 7 Alexis took a total of 243 000 steps during the 30 days in the month of April. What was her mean (average) number of steps per day in April? (B) 8100 (C) 8000 (D) 7100 (E) 8200 In the pie chart shown, 80 students chose juice. How many students chose milk? (A) 120 (B) 160 (D) 180 (E) 80 juice (C) 240 milk In an increasing list of consecutive integers, the 3rd and 4th numbers in the list add to 11. What is the 6th number in the list? (A) 10 8. (E) $20 y In the morning, the temperature was −3◦ C. In the afternoon, the temperature was 5◦ C. By how many degrees Celsius did the temperature increase? (A) 7900 6. (D) $2 Which of the following integers is closest to 99 × 9? (A) 8 5. (C) $12 In the diagram shown, what are the coordinates of point P ? (A) 10 000 4. (B) $10 (B) 11 (C) 9 (D) 8 Tick marks are equally spaced along a number line with the numbers P , Q, R, S, T , and U labelled as shown. Which of the following best represents the value of R divided by the value of U ? (A) 0.25 (B) 0.50 (D) 1.25 (E) 1.50 (E) 12 P Q R S (B) 10 (D) 14 (E) 15 U 1.0 0 (C) 0.75 14 cm In the diagram, the perimeter of the triangle is equal to the perimeter of the rectangle. What is the value of x? (A) 8 T 12 cm (C) 11 8 cm x cm 10. The positive divisors of 12 (other than itself) are 1, 2, 3, 4, and 6. Their sum, 1 + 2 + 3 + 4 + 6, is greater than 12. An abundant number is a number for which the sum of its positive divisors (other than itself) is greater than the number itself. This means that 12 is an abundant number. Which of the following is also an abundant number? (A) 8 (B) 10 (C) 14 (D) 18 (E) 22 Grade 7 Part B: Each correct answer is worth 6. 11. Each of 7 boxes contains exactly 10 cookies. If the cookies are shared equally among 5 people, how many cookies does each person receive? (A) 14 (B) 12 (C) 9 (D) 11 (E) 13 12. Abdul is 9 years older than Susie, and Binh is 2 years older than Susie. How many years older is Abdul than Binh? (A) 11 (B) 9 (C) 14 (D) 2 13. Points P (15, 55), Q(26, 55) and R(26, 35) are three vertices of rectangle P QRS. The area of this rectangle is (A) 360 (B) 800 (D) 580 (E) 330 (E) 7 P (15, 55) Q (26, 55) S R (26, 35) (C) 220 14. A box contains 15 red, 20 blue, and 16 green jelly beans. Jack first chooses a green jelly bean and eats it. Then he chooses a blue jelly bean and eats it. If each of the remaining jelly beans is equally likely to be chosen, what is the probability that Jack chooses a red jelly bean next? (A) 15 31 (B) 34 49 (C) 15 49 (D) 2 7 (E) 1 15 15. Emil and Olivia ran a race. Their race times totalled 1 hour 52 minutes. If Emil’s time was 4 minutes less than Olivia’s time, how many minutes did it take Olivia to run the race? (A) 78 (B) 56 (C) 58 (D) 74 16. In the diagram, which of the following squares should be shaded to make BD a line of symmetry of square ABCD? (A) P and S (B) Q and S (C) P and T (D) Q and T (E) 55 P Q R D C S T (E) P and R A B 17. Rosie is saving money. She has $120 in her account today and will begin saving by making $30 deposits into her account. If she makes m such deposits, the expression that best represents the number of dollars in her account is (A) 120 + m (B) 30m (C) 30 + 120m (D) 150m (E) 120 + 30m 18. Two isosceles triangles each have at least one angle that measures 70◦. In the first triangle, the measure in degrees of each of the remaining two angles is even. In the second triangle, the measure in degrees of each of the remaining two angles is odd. In the first triangle, the sum of the equal angles is S. In the second triangle, the sum of the equal angles is T. The value of S + T is (A) 280◦ (B) 250◦ (C) 220◦ (D) 200◦ 19. Three different views of the same cube are shown. The symbol on the face opposite is (A) (B) (D) (E) (C) (E) 300◦ Grade 7 20. On the grid shown, Jane starts at dot A. She tosses a fair coin to determine which way to move. If she tosses a head, she moves up one dot. If she tosses a tail she moves right one dot. After four tosses of the coin, Jane will be at one of the dots P, Q, R, S, or T. What is the probability that Jane will be at dot R? (A) (D) 1 2 7 16 (B) (E) 3 8 5 16 (C) P Q R S 9 16 T A Part C: Each correct answer is worth 8. 21. A four-digit number can be made by repeating a two-digit number. For example, 1111 is made by repeating 11, and 1919 is made by repeating 19. How many such numbers are there between 2000 and 10 000? (A) 80 (B) 81 (C) 79 (D) 72 (E) 70 22. Celyna bought 300 grams of candy A for $5.00, and x grams of candy B for $7.00. She calculated that the average price of all of the candy that she purchased was $1.50 per 100 grams. What is the value of x? (A) 525 (B) 600 (C) 500 (D) 450 (E) 900 23. The list 11, 20, 31, 51, 82 is an example of an increasing list of five positive integers in which the first and second integers add to the third, the second and third add to the fourth, and the third and fourth add to the fifth. How many such lists of five positive integers have 124 as the fifth integer? (A) 10 (B) 7 (C) 9 (D) 6 (E) 8 F 24. In the 10 by 10 grid of squares shown, point P can be at any of the 41 points of intersection of pairs of gridlines inside (and not on) 4F GH. For each possible location of P , exactly three triangles are formed: 4F P G, 4GP H, 4HP F. How many of these 123 triangles have an area that is exactly half of the area of 4F GH? (A) 5 (B) 7 (D) 11 (E) 9 (C) 3 25. Every 12 minutes, Bus A completes a trip from P to X to S to X to P. Every 20 minutes, Bus B completes a trip from Q to X to T to X to Q. Every 28 minutes, Bus C completes a trip from R to X to U to X to R. At 1:00 p.m., Buses A, B and C depart from P , Q and R, respectively, each driving at a constant speed, and each turning around instantly at the endpoint of its route. Each bus runs until 11:00 p.m. At how many times between 5:00 p.m. and 10:00 p.m. will two or more buses arrive at X at the same time? (A) 18 (B) 19 (C) 20 (D) 21 (E) 22 G H U P Q R X S T 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 15, 2019 (in North America and South America) Thursday, May 16, 2019 (outside of North America and South America) Time: 1 hour ©2018 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. Erin receives $3 a day. How many days will it take Erin to receive a total of $30? (A) 8 (B) 12 (C) 14 (D) 27 (E) 10 y 3. 4. In the diagram, point F has coordinates (5, 5). The point with coordinates (2, 4) is located at (A) A (B) B (D) D (E) E (B) 40% (D) 50% (E) 15% C A E x P Q S R (C) 25% The value of 0.9 + 0.09 is (A) 1.08 5. B In the diagram, square P QRS is divided into four identical squares. What percentage of square P QRS is shaded? (A) 33% F (5, 5) D (C) C (B) 0.909 (C) 1.8 (D) 0.99 Based on the graph shown, what is the mode for the amount of rainfall for the week? (A) 9 mm (B) 12 mm (D) 15 mm (E) 6 mm (C) 3 mm Amount of Rainfall (mm) 2. 15 12 9 6 3 0 (E) 0.18 Rainfall in Gaussburg Su M Tu W Th F Sa Day of the Week 6. If x = 3, which of the following is true? (A) 2x = 5 7. When two numbers are added, the result is −26. If one of the numbers is 11, what is the other number? (A) −37 8. (B) 37 (C) −15 (D) 15 (E) −48 Joshua is reading a 396-page book. He has read the first third of the book only. How many pages does he have left to read to finish the rest of the book? (A) 264 9. (B) 3x − 1 = 8 (C) x + 5 = 3 (D) 7 − x = 2 (E) 6 + 2x = 14 (B) 124 (C) 250 (B) 210 (D) 270 (E) 300 (E) 244 k˚ In the diagram, the value of k is (A) 180 (D) 199 (C) 240 Grade 7 10. The mean (average) of the numbers 20, 30, 40 is equal to the mean of the numbers (A) 28, 30, 31 (D) 23, 30, 37 (B) 24, 30, 38 (E) 25, 30, 34 (C) 22, 30, 39 Part B: Each correct answer is worth 6. √ 11. The value of 81 is equal to (A) 3 (B) 32 (C) 33 12. In the diagram, what is the area of rectangle P QRS? (A) 36 (B) 32 (D) 20 (E) 44 (D) 34 (E) 35 y S (– 4, 2) R (4, 2) x (C) 40 P (– 4, – 2) Q (4, – 2) 13. A piano has 52 white keys that occur in a repeating pattern of ABCDEFG. The first white key is A. What letter is associated with the 33rd white key? (A) A (B) B (C) C (D) D 14. A circular spinner is divided into 8 equal sections, as shown. 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 section containing a prime number that is odd? 1 2 3 (A) (B) (C) 8 8 8 4 7 (D) (E) 8 8 (E) E 9 2 8 3 7 4 6 5 15. Canadian currency has coins with values $2.00, $1.00, $0.25, $0.10, and $0.05. Barry has 12 coins including at least one of each of these coins. What is the smallest total amount of money that Barry could have? (A) $3.75 (B) $3.90 (C) $3.70 (D) $3.40 (E) $3.95 16. A positive integer whose digits are the same when read forwards or backwards is called a palindrome. For example 474 and 222 are palindromes. How many palindromes are there between 100 and 1000? (A) 10 (B) 90 (C) 100 (D) 900 (E) 1000 17. The two equal-arm scales shown are balanced. Of the following, has the same mass as (A) (B) (C) (D) (E) 18. A rectangle has length x and width y. A triangle has base 16 and height x. If the area of the rectangle is equal to the area of the triangle, then the value of y is (A) 16 (B) 4 (C) 8 (D) 12 (E) 32 Grade 7 19. Each of a, b, c, and d is a positive integer and is greater than 3. If 1 1 1 1 = = = a−2 b+2 c+1 d−3 then which ordering of these four numbers is correct? (A) a < b < c < d (B) c < b < a < d (D) d < a < c < b (E) b < c < a < d (C) b < a < c < d 20. The positive integer n has exactly 8 positive divisors including 1 and n. Two of these divisors are 14 and 21. What is the sum of all 8 positive divisors of n? (A) 35 (B) 47 (C) 53 (D) 96 (E) 103 Part C: Each correct answer is worth 8. 21. Kathy owns more cats than Alice and more dogs than Bruce. Alice owns more dogs than Kathy and fewer cats than Bruce. Which of the statements must be true? (A) Bruce owns the fewest cats. (B) Bruce owns the most cats. (C) Kathy owns the most cats. (D) Alice owns the most dogs. (E) Kathy owns the fewest dogs. 22. Each of the integers 334 and 419 has digits whose product is 36. How many 3-digit positive integers have digits whose product is 36? (A) 21 (B) 15 (C) 18 (D) 24 (E) 12 23. Points T, U, V, W, X, Y lie on square P QRS, as shown. If P T = T U = U Q = QV = V W = W R = XS = SY , what fraction of the area of square P QRS is shaded? 5 1 2 (A) (B) (C) 18 3 9 1 1 (D) (E) 4 6 P T U Q V Y S W X R 24. A dot starts at (20, 19). It can move one unit vertically or horizontally to one of the points (21, 19), (19, 19), (20, 20), or (20, 18). From there it can move two units in either direction that is perpendicular to the first move. All moves thereafter increase in length by one unit (three units, four units, five units, etc.) and must be perpendicular to the direction of the previous move. The dot stops after ten moves. Which of the following final locations is not possible? (A) (27, 33) (B) (30, 40) (C) (21, 21) (D) (42, 44) (E) (37, 37) 25. An 8 × 8 × n rectangular prism is made up from 1 × 1 × 1 cubes. Suppose that A is the surface area of the prism and B is the combined surface area of the 1 × 1 × 1 B cubes that make up the prism. What is the sum of the values of n for which is an A integer? (A) 86 (B) 90 (C) 70 (D) 78 (E) 96 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 16, 2018 (in North America and South America) Thursday, May 17, 2018 (outside of North America and South America) Time: 1 hour ©2017 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) previously stored information 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. What number should be subtracted from 21 to give 8? (A) 12 2. (B) 80 (E) 16 Banana 20% Orange 40% Apple 40% (C) 100 (B) 25 (C) 35 (D) 45 (E) 75 A square has an area of 144 cm2. The side length of the square is (A) 288 cm 5. (D) 15 (D) 20 (E) 60 A class begins at 8:30 a.m. and ends at 9:05 a.m. on the same day. In minutes, what is the length of the class? (A) 15 4. (C) 14 In the diagram, the pie chart shows the results of a survey asking students to choose their favourite fruit. 100 students were surveyed. How many students chose banana? (A) 40 3. (B) 13 (B) 72 cm (C) 48 cm (D) 12 cm (E) 36 cm If there is no tax, which of the following costs more than $18 to purchase? (A) Five $1 items and five $2 items (B) Nine $1 items and four $2 items (C) Nine $1 items and five $2 items (D) Two $1 items and six $2 items (E) Sixteen $1 items and no $2 items 6. Which of the following numbers lies between 3 and 4 on a number line? (A) 7. 2 9 11 4 (C) 11 5 (D) 13 4 (E) 13 5 (B) 5 9 (C) 9 7 (D) 7 9 (E) 1 9 If x = 4 and y = 3x, the value of y is (A) 12 9. (B) An envelope contains 2 sunflower seeds, 3 green bean seeds, and 4 pumpkin seeds. Carrie randomly chooses one of the seeds from the envelope. What is the probability that Carrie chooses a sunflower seed? (A) 8. 5 2 (B) 24 (C) 7 (D) 81 (E) 4 The measure of one angle of an isosceles triangle is 50◦. The measures of the other angles in this triangle could be (A) 50◦ and 90◦ (D) 30◦ and 100◦ (B) 40◦ and 50◦ (E) 60◦ and 70◦ (C) 50◦ and 80◦ 10. The 26 letters of the alphabet are written in order, clockwise around a circle. The ciphertext of a message is created by replacing each letter of the message by the letter that is 4 letters clockwise from the original letter. (This is called a Caesar cipher.) For example, the message ZAP has ciphertext DET. What is the ciphertext of the message W IN ? (A) ALN (B) ZLN (C) AM R (D) AM Q (E) ZM Q Grade 7 Part B: Each correct answer is worth 6. 11. A cube has exactly six faces and twelve edges. How many vertices does a cube have? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 12. What is the surface area of a 1 cm by 2 cm by 2 cm rectangular prism? (A) 10 cm2 (B) 20 cm2 (D) 24 cm2 (E) 16 cm2 1 cm (C) 12 cm2 2 cm 2 cm 13. At a factory, 11 410 kg of rice is distributed equally into 3260 bags. A family uses 0.25 kg of rice each day. How many days would it take this family to use up one bag of rice? (A) 9 (B) 12 (C) 13 (D) 14 (E) 15 14. Dalia’s birthday is on a Wednesday and Bruce’s birthday is 60 days after Dalia’s. On what day of the week is Bruce’s birthday? (A) Monday (B) Tuesday (C) Friday (D) Saturday (E) Sunday 15. Karl has 30 birds. Some of his birds are emus and the rest are chickens. Karl hands out 100 treats to his birds. Each emu gets 2 treats and each chicken gets 4 treats. How many chickens does Karl have? (A) 10 (B) 15 (C) 25 (D) 20 (E) 6 16. The integers 1 to 32 are spaced evenly and in order around the outside of a circle. Straight lines that pass through the centre of the circle join these numbers in pairs. Which number is paired with 12? (A) 28 (B) 27 (C) 23 (D) 21 (E) 29 17. In the diagram, the area of the shaded middle ring is 6 times the area of the smallest circle. The area of the unshaded outer ring is 12 times the area of the smallest circle. What fraction of the area of the largest circle is the area of the smallest circle? (A) (D) 1 3 1 18 (B) (E) 1 6 1 19 (C) 1 12 18. There are several groups of six integers whose product is 1. Which of the following cannot be the sum of such a group of six integers? (A) −6 (B) −2 (C) 0 (D) 2 (E) 6 19. The heights of 4 athletes on a team are 135 cm, 160 cm, 170 cm, and 175 cm. Laurissa joins the team. On the new team of 5 athletes, the mode height of the players is equal to the median height which is equal to the mean (average) height. How tall is Laurissa? (A) 135 cm (B) 160 cm (C) 170 cm (D) 175 cm (E) 148 cm Grade 7 20. In the diagram, 4P QR is isosceles with P Q = P R. What is the value of x? (A) 110 (B) 90 (D) 100 (E) 105 P x° (C) 95 120° 95° Q R Part C: Each correct answer is worth 8. 21. The figure consists of 8 identical small parallelograms, joined as shown. Including these 8 small parallelograms, how many parallelograms appear in this figure? (A) 29 (B) 30 (D) 27 (E) 28 (C) 26 22. In a jar, there are 50 coins with a total value of $5.00. The coins are quarters (worth $0.25 each), dimes (worth $0.10 each), and nickels (worth $0.05 each). The number of nickels in the jar is three times the number of quarters. The number of dimes is one more than the number of nickels. How many quarters are in the jar? (A) 7 (B) 6 (C) 5 (D) 9 (E) 8 23. The digits from 1 to 9 are written in order so that the digit n is written n times. This forms the block of digits 1223334444 · · · 999999999. The block is written 100 times. What is the 1953rd digit written? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 24. The number 2018 is used to create six-digit positive integers. These six-digit integers must contain the digits 2018 together and in this order. For example, 720 186 is allowed, but 209 318 and 210 893 are not. How many of these six-digit integers are divisible by 9? (A) 28 (B) 27 (C) 31 (D) 34 25. In the triangle, each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 is placed into a different circle. The sums of the numbers on each of the three sides of the triangle are equal to the same number, S. The sum of all of the different possible values of S is (A) 85 (B) 99 (D) 81 (E) 67 (C) 66 (E) 22 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 10, 2017 (in North America and South America) Thursday, May 11, 2017 (outside of North America and South America) Time: 1 hour ©2016 University of Waterloo Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. The value of (2 + 4 + 6) − (1 + 3 + 5) is (D) 21 (B) basketball (C) soccer (D) volleyball (E) badminton 3. 7. (B) 5 (B) 0.3 (A) 0 (B) 45 (D) 180 (E) 360 (E) 18 (C) 7 (D) 9 (E) 33 (C) 0.03 (D) 30 (E) 0.003 (C) 60 Which integer is closest in value to (B) 8 xº 35 4 ? (C) 9 (D) 7 (E) 6 When n = 101, which of the following expressions has an even value? (A) 3n 9. (D) 16 In the square shown, x is equal to (A) 10 8. (C) 14 Three thousandths is equal to (A) 300 6. (B) 12 When two integers between 1 and 10 are multiplied, the result is 14. What is the sum of these two integers? (A) 2 5. Sport Michael has $280 in $20 bills. How many $20 bills does he have? (A) 10 4. 50 40 30 20 10 0 badminton (A) hockey Sports Played by Students soccer Based on the graph shown, which sport is played by the most students? (E) 111 volleyball (C) −3 basketball 2. (B) 3 hockey (A) 0 Number of Students 1. (B) n + 2 (C) n − 12 (D) 2n − 2 (E) 3n + 2 The sum of three consecutive integers is 153. The largest of these three integers is (A) 52 (B) 50 (C) 53 (D) 54 (E) 51 10. In the diagram, 4P QR is equilateral and is made up of four smaller equilateral triangles. If each of the smaller triangles has a perimeter of 9 cm, what is the perimeter of 4P QR? (A) 15 cm (B) 9 cm (D) 27 cm (E) 18 cm P (C) 36 cm Q R Grade 7 Part B: Each correct answer is worth 6. 11. The number that goes into the  to make (A) 27 (B) 9 (C) 59 3  = true is 7 63 (D) 63 (E) 3 12. At the Gaussian Store, puzzles cost $10 each or $50 for a box of 6 puzzles. If a customer would like exactly 25 puzzles, what is the minimum possible cost? (A) $210 (B) $230 (C) $250 (D) $220 (E) $200 13. When the shaded triangle shown is translated, which of the following triangles can be obtained? (A) A (B) B (D) D (E) E (C) C y A D E x C B 14. When the time in Toronto, ON is 1:00 p.m., the time in Gander, NL is 2:30 p.m. A flight from Toronto to Gander takes 2 hours and 50 minutes. If the flight departs at 3:00 p.m. (Toronto time), what time will the flight land in Gander (Gander time)? (A) 7:20 p.m. (B) 5:00 p.m. (C) 6:20 p.m. (D) 5:20 p.m. (E) 8:50 p.m. 15. Five students ran a race. Ryan was faster than Henry and Faiz. Henry was slower than Faiz. Toma was faster than Ryan but slower than Omar. Which student finished fourth? (A) Faiz (B) Henry (C) Omar (D) Ryan (E) Toma 16. A circular spinner is divided into 20 equal sections, as shown. 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 section containing a number that is a divisor of 20? (A) (D) 12 20 7 20 (B) (E) 14 20 6 20 (C) 15 20 19 20 1 2 3 18 4 17 5 16 15 6 14 7 13 8 12 11 10 9 17. The mean (average) of the four integers 78, 83, 82, and x is 80. Which one of the following statements is true? (A) x is 2 greater than the mean (B) x is 1 less than the mean (C) x is 2 less than the mean (D) x is 3 less than the mean (E) x is equal to the mean 18. Sara goes to a bookstore and wants to buy a book that is originally priced at $100. Which of the following options gives her the best discounted price? (A) A discount of 20% (B) A discount of 10%, then a discount of 10% off the new price (C) A discount of 15%, then a discount of 5% off the new price (D) A discount of 5%, then a discount of 15% off the new price (E) All four options above give the same price Grade 7 19. Two sheets of 11 cm × 8 cm paper are placed on top of each other, forming an overlapping 8 cm × 8 cm square in the centre, as shown. The area of rectangle W XY Z is (A) 88 cm2 (B) 112 cm2 (C) 136 cm2 (D) 121 cm2 (E) 176 cm2 20. Betty and Ann are walking around a rectangular park with dimensions 600 m by 400 m, as shown. They both begin at the top left corner of the park and walk at constant but different speeds. Betty walks in a clockwise direction and Ann walks in a counterclockwise direction. Points P, Q, R, S, T divide the bottom edge of the park into six segments of equal length. When Betty and Ann meet for the first time, they are between Q and R. Which of the following could be the ratio of Betty’s speed to Ann’s speed? (A) 5 : 3 (B) 9 : 4 (D) 12 : 5 (E) 17 : 7 W X Z Y Ann (D) 84 (E) 100 4 2 (C) 90 22. In the six-digit number 1ABCDE, each letter represents a digit. 1ABCDE × 3 = ABCDE1, the value of A + B + C + D + E is (A) 29 (B) 26 y (C) 11 : 6 21. Rectangles that measure 4 × 2 are positioned in a pattern in which the top left vertex of each rectangle (after the top one) is placed at the midpoint of the bottom edge of the rectangle above it, as shown. When a total of ten rectangles are arranged in this pattern, what is the perimeter of the figure? (B) 64 400 m P Q R S T 600 m Part C: Each correct answer is worth 8. (A) 48 Betty Start (C) 22 (D) 30 Given that (E) 28 23. Given 8 dimes (10 ¢ coins) and 3 quarters (25 ¢ coins), how many different amounts of money can be created using one or more of the 11 coins? (A) 27 (B) 29 (C) 35 (D) 26 (E) 28 24. Four vertices of a quadrilateral are located at (7, 6), (−5, 1), (−2, −3), and (10, 2). The area of the quadrilateral in square units is (A) 60 (B) 63 (C) 67 (D) 70 (E) 72 25. Ashley writes out the first 2017 positive integers. She then underlines any of the 2017 integers that is a multiple of 2, and then underlines any of the 2017 integers that is a multiple of 3, and then underlines any of the 2017 integers that is a multiple of 5. Finally, Ashley finds the sum of all the integers which have not been underlined. What is this sum? (A) 542 708 (B) 543 213 (C) 542 203 (D) 543 326 (E) 543 618 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 11, 2016 (in North America and South America) Thursday, May 12, 2016 (outside of North America and South America) Time: 1 hour ©2015 University of Waterloo Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. The value of 333 + 33 + 3 is (A) 396 2. (A) Friday (B) Tuesday (D) Saturday (E) Wednesday Saturday Sunday Friday Thursday (C) 77 Wednesday (B) 76 (B) 3 7 (B) (D) 78 (E) 79 (C) 4 7 (D) 3 8 (E) 4 9 (C) (D) (E) The measures of two angles of a triangle are 25◦ and 70◦. The measure of the third angle is (B) 105◦ (C) 65◦ (D) 95◦ (E) 75◦ A box of fruit contains 20 apples, 10 oranges, and no other fruit. When a fruit is randomly chosen from the box, what is the probability that the fruit is an orange? 1 10 (B) 1 20 (C) 1 30 (D) 1 3 (E) 2 3 Alex pays $2.25 to take the bus. Sam pays $3.00 to take the bus. If they each take the bus 20 times, how much less would Alex pay than Sam in total? (A) $25 9. Day of the Week A cube has exactly one face painted as shown. The other five faces of the cube are not painted. If the cube is rolled, which of the following could be the same cube? (A) 8. 50 40 30 20 10 0 Tuesday 2 5 (A) 85◦ 7. Text Messages Received by Tanner Which of these fractions is larger than 21 ? (A) 6. (C) Thursday (E) 963 Which of the following is a multiple of 7? (A) 5. (D) 369 The graph shows the number of text messages received by Tanner in a given week. On what day did Tanner receive the most text messages? (A) 75 4. (C) 669 Monday 3. (B) 399 Number of Messages 1. (B) $10 (C) $15 (D) $45 (E) $60 Carrie is travelling at a constant speed of 85 km/h. If Carrie is halfway through a 510 km trip, how much longer will the trip take? (A) 5 hours (B) 425 hours (C) 12 hours (D) 1.5 hours (E) 3 hours 10. Points P, Q and R are on a number line. Q is halfway between P and R. If P is at −6 and Q is at −1, then R is at (A) 4 (B) −11 (C) 3 (D) −7 (E) 5 Grade 7 Part B: Each correct answer is worth 6. 11. The diagram shown contains octagons and squares only. The ratio of the number of octagons to the number of squares is (A) 1 : 1 (B) 2 : 1 (D) 5 : 4 (E) 5 : 3 (C) 25 : 12 P QQ 12. In the sum shown, P and Q each represent a digit. The value of P + Q is (A) 3 (B) 5 (D) 6 (E) 4 PPQ (C) 7 + QQQ 876 13. A larger cube has volume 64 cm3. A smaller cube has edges that are half the length of the edges of the larger cube. What is the volume of the smaller cube? (A) 24 cm3 (B) 48 cm3 (C) 8 cm3 (D) 16 cm3 (E) 27 cm3 14. Ahmed chooses two different items for a snack. His choices are an apple, an orange, a banana, and a granola bar. How many different pairs of snacks could he choose? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 15. Sophia did push-ups every day for 7 days. Each day after the first day, she did 5 more push-ups than the day before. In total she did 175 push-ups. How many push-ups did Sophia do on the last day? (A) 55 (B) 35 (C) 50 (D) 45 (E) 40 16. Each of , 4 and  represents a non-zero number. If  = 4 + 4 + 4 and  =  + , then  +  + 4 equals (A)  + 4 (D) 4 + 4 + 4 +  +  (B)  + 4 + 4 + 4 + 4 (E)  +  +  + 4 + 4 (C)  +  +  y D 17. Triangle T is reflected once. Which of the following triangles cannot be this reflection of triangle T ? (A) A (B) B (D) D (E) E A (C) C B T x C E 18. The mean (average) of a set of six numbers is 10. When the number 25 is removed from the set, the mean of the remaining numbers is (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 19. Suzy’s 5 m long ribbon has shaded and unshaded sections of equal length, as shown. Points A, B, C, D, E are equally spaced along the ribbon. If Suzy wants a ribbon that is make a single vertical cut? (A) A (B) B 11 15 A B C D E of the size of this ribbon, at which point could she (C) C (D) D (E) E Grade 7 20. In the diagram, four different integers from 1 to 9 inclusive are placed in the four boxes in the top row. The integers in the left two boxes are multiplied and the integers in the right two boxes are added and these results are then divided, as shown. The final result is placed in the bottom box. Which of the following integers cannot appear in the bottom box? (A) 16 (B) 24 (D) 20 (E) 9 (C) 7 Part C: Each correct answer is worth 8. 21. A 10 by 10 grid is created using 100 points, as shown. Point P is given. One of the other 99 points is randomly chosen to be Q. What is the probability that the line segment P Q is vertical or horizontal? (A) (D) 2 11 4 25 (B) (E) 1 5 5 33 (C) P 1 10 22. The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: (1, 2, 5, 8), (3, 4, 6, 7), (2, 4, 5, 7), (1, 3, 6, 8), (2, 3, 7, 8), and (1, 4, 5, 6). The label of the vertex of the cube that is farthest away from the vertex labelled 2 is (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 23. Angie has a jar that contains 2 red marbles, 2 blue marbles, and no other marbles. She randomly draws 2 marbles from the jar. If the marbles are the same colour, she discards one and puts the other back into the jar. If the marbles are different colours, she discards the red marble and puts the blue marble back into the jar. She repeats this process a total of three times. What is the probability that the remaining marble is red? (A) 21 (B) 41 (C) 23 (D) 13 (E) 0 24. How many of the five numbers 101, 148, 200, 512, 621 cannot be expressed as the sum of two or more consecutive positive integers? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 25. In the triangle shown, the first diagonal line, 1, 2, 3, 4,... , begins at 1 and each number after the first is one larger than the previous number. The second diagonal line, 2, 4, 6, 8,... begins at 2 and each number after the first is two larger than the previous number. The nth diagonal line begins at n and each number after the first is n larger than the previous number. In which horizontal row does the number 2016 first appear? (A) 90 (B) 94 (D) 91 (E) 89 1 2 3 4 5 (C) 88 6 4 6 8 10 2 3 6 9 12 4 8 12 5 10 6 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 13, 2015 (in North America and South America) Thursday, May 14, 2015 (outside of North America and South America) Time: 1 hour ©2014 University of Waterloo Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. In the diagram, the fraction of the circle that is shaded is equal to (A) (D) 2. 1 2 1 6 (E) 1 3 1 8 (C) 1 4 The value of 10 × (5 − 2) is (A) 13 3. (B) (B) 70 (C) 7 (D) 30 The graph shows the total distance that each of five runners ran during a one-hour training session. Which runner ran the least distance? (A) Phil (B) Tom (D) Amal (E) Sanjay (C) Pete (E) 50 Total Distance Run 8 Distance (km) 1. 6 4 2 0 Sanjay Runners Amal Pete The equal-arm scale shown is balanced. One has the same mass as Tom Phil 4. (A) (B) (C) (D) (E) 5. Which of the following is closest to 5 cm? (A) The length of a full size school bus (B) The height of a picnic table (C) The height of an elephant (D) The length of your foot (E) The length of your thumb 6. The number of centimetres in 3.5 metres is (A) 350 7. (B) 30.5 (C) 3.05 (D) 3.50 (E) 305 The perimeter of the figure shown is (A) 18 (B) 17 (D) 20 (E) 25 (C) 23 5 3 2 3 2 8. Hannah scored 312 points during the basketball season. If her average (mean) was 13 points per game, how many games did she play? (A) 24 9. (B) 41 (C) 17 (D) 13 (E) 30 The number 6 has exactly four positive divisors: 1, 2, 3, and 6. How many positive divisors does 20 have? (A) 2 (B) 6 (C) 3 (D) 5 (E) 8 Grade 7 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? (A) 6 (B) 3 (C) 5 (D) 12 (E) 9 Art Part B: Each correct answer is worth 6. 10% 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? (A) 184 (B) 192 (C) 96 (D) 144 Math Music 40% 20% Science 30% (E) 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 (A) 16 (B) 32 (C) 25 (D) 8 (E) 64 13. How many even whole numbers between 1 and 99 are multiples of 5? (A) 5 (B) 7 (C) 9 (D) 11 (E) 13 14. In the 3 × 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 (A) 3 (B) 2 (D) 6 (E) 4 1 X 3 (C) 5 Y 2 cm 15. In the rectangle shown, the area of the shaded region is (A) 60 cm2 (B) 20 cm2 (D) 40 cm2 (E) 50 cm2 (C) 30 cm2 5 cm 12 cm 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? (A) 20 (B) 11 (C) 10 (D) 12 (E) 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? (A) 18 (B) 14 (D) 15 (E) 13 (C) 24 18. In the graph shown, which of the following represents the image of the line segment P Q after a reflection across the x-axis? (A) P S (B) T U (D) W V (E) F G y Q N M (C) M N F G P T S V W U x Grade 7 19. When expressed as a repeating decimal, the fraction 17 is written as 0.142857142857... (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? (A) 119th (B) 121st (C) 123rd (D) 125th (E) 126th 20. In a triangle, the measure of one of the angles is 45◦. 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? (A) 80◦ (B) 90◦ (C) 75◦ (D) 85◦ (E) 100◦ 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 10 11 20 21 2 9 12 19 22 3 8 13 18 23 4 7 14 17 24 5 6 15 16 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? (A) 75 (B) 73 (C) 71 (D) 70 (E) 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? (A) P (B) 2P (C) 21 P (D) 54 P (E) 25 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 (A) 4 : 5 (B) 5 : 2 (C) 2 : 7 (D) 4 : 3 24. In the diagram, rectangle P QRS is made up of six identical squares. Points U, V, W, X, Y, and Z are midpoints of sides of the squares, as shown. Which of the following triangles has the greatest area? (A) P V U (B) P XZ (D) P Y S (E) P QW (E) 1 : 10 Q V W R X U (C) P V X P Y Z S 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? (A) 18 (B) 14 (C) 20 (D) 17 (E) 19 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 14, 2014 (in North America and South America) Thursday, May 15, 2014 (outside of North America and South America) Time: 1 hour ©2014 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 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: 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. 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. The name, school and location of some top-scoring students will be published on the website, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. The value of (4 × 3) + 2 is (A) 33 2. (D) 3 6 2 6 3 4 (E) 110 (B) 10 (C) 15 (D) 25 (B) (E) 4 6 1 6 (C) 5 6 (E) 20 P R Q P R Q (B) 16 (C) 8 (B) 2 3 (C) (D) 64 (E) 32 15 25 ? 3 5 (D) 1 2 (E) 5 7 (B) 9 (C) 15 (D) 12 (E) 13 If 9210 − 9124 = 210 − , the value represented by the  is (A) 296 9. (D) 107 How many positive two-digit whole numbers are divisible by 7? (A) 11 8. (C) 103 Which of these fractions is equivalent to (A) 7. (E) 11 One scoop of fish food can feed 8 goldfish. How many goldfish can 4 scoops of fish food feed? (A) 12 6. (B) 95 The spinner shown is divided into 6 sections of equal size. What is the probability of landing on a section that contains the letter Q using this spinner? (A) 5. (D) 24 Five times a number equals one hundred. The number is (A) 50 4. (C) 14 Which of the following numbers is closest to 100 on the number line? (A) 98 3. (B) 10 (B) 210 (C) 186 (D) 124 (E) 24 A clockwise rotation around point Z (that is, a rotation in the direction of the arrow) transforms the shaded quadrilateral to the unshaded quadrilateral. The angle of rotation is approximately (A) 180◦ (B) 270◦ (D) 45◦ (E) 135◦ (C) 360◦ Z 10. Which one of the following is equal to 17? (A) 3 − 4 × 5 + 6 (D) 3 ÷ 4 + 5 − 6 (B) 3 × 4 + 5 ÷ 6 (E) 3 × 4 ÷ 5 + 6 (C) 3 + 4 × 5 − 6 Part B: Each correct answer is worth 6. 11. Consider the set {0.34, 0.304, 0.034, 0.43}. The sum of the smallest and largest numbers in the set is (A) 0.77 (B) 0.734 (C) 0.077 (D) 0.464 (E) 0.338 Grade 7 12. The diagonals have been drawn in the square shown. The area of the shaded region of the square is (A) 4 cm2 (B) 8 cm2 (D) 56 cm2 (E) 64 cm2 (C) 16 cm2 13. In the special square shown, the sum of the three numbers in each column equals the sum of the three numbers in each row. The value of x is (A) 3 (B) 4 (D) 6 (E) 12 (C) 5 13 8 14 x 10 9 14. In the diagram shown, the number of rectangles of all sizes is (A) 11 (B) 15 (D) 13 (E) 9 (C) 7 15. The diagram shows Lori’s house located at (6, 3). If Alex’s house is located at (−2, −4), what translation is needed to get from Lori’s house to Alex’s house? y Lori’s House (6, 3) (A) 4 units left, 1 unit up (B) 8 units right, 7 units up (C) 4 units left, 1 unit down (D) 8 units left, 7 units down (E) 7 units right, 8 units down x 16. The graph shows points scored by Riley-Ann in her first five basketball games. The difference between the mean and the median of the number of points that she scored is (A) 1 (B) 2 (C) 3 (D) 4 Riley-Ann’s Points Points (E) 5 17. In the diagram shown, P QR is a straight line segment. The measure of ∠QSR is (A) 25◦ (B) 30◦ (D) 40◦ (E) 45◦ 18. In the figure shown, the outer square has an area of 9 cm2 , the inner square has an area of 1 cm2 , and the Lori’s House four rectangles are identical. What is the perimeter of one of the four identical rectangles? (B) 8 cm (D) 9 cm (E) 7 cm 1 2 3 Game 4 (C) 35◦ P (A) 6 cm S 22 20 18 16 14 12 10 8 6 4 2 0 (C) 10 cm 75° Q 30° R 5 Grade 7 19. Sarah’s hand length is 20 cm. She measures the dimensions of her rectangular floor to be 18 by 22 hand lengths. Which of the following is the closest to the area of the floor? (A) 160 000 cm2 (B) 80 000 cm2 (C) 200 000 cm2 2 2 (D) 16 000 cm (E) 20 000 cm 20. The product of three consecutive odd numbers is 9177. What is the sum of the numbers? (A) 51 (B) 57 (C) 60 (D) 63 (E) 69 Part C: Each correct answer is worth 8. 21. A bicycle at Store P costs $200. The regular price of the same bicycle at Store Q is 15% more than it is at Store P. The bicycle is on sale at Store Q for 10% off of the regular price. What is the sale price of the bicycle at Store Q? (A) $230.00 (B) $201.50 (C) $199.00 (D) $207.00 (E) $210.00 22. Each face of a cube is painted with exactly one colour. What is the smallest number of colours needed to paint a cube so that no two faces that share an edge are the same colour? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 23. Two standard six-sided dice are tossed. One die is red and the other die is blue. What is the probability that the number appearing on the red die is greater than the number appearing on the blue die? 25 (B) 36 (C) 15 (D) 12 (E) 17 (A) 18 36 36 36 36 24. In the diagram shown, Q S ST U V is a square, T Q and P are the midpoints of ST and U V , R P R = QR, and V Q is parallel to P R. V What is the ratio of the shaded area to the unshaded area? (A) 2 : 3 (B) 3 : 5 (D) 7 : 9 (E) 5 : 7 U (C) 1 : 1 25. On a coordinate grid, Paul draws a line segment of length 1 from the origin to the right, stopping at (1, 0). He then draws a line segment of length 2 up from this point, stopping at (1, 2). He continues to draw line segments to the right and up, increasing the length of the line segment he draws by 1 each time. One of his line segments stops at the point (529, 506). What is the endpoint of the next line segment that he draws? (A) (529, 552) (B) (576, 506) (C) (575, 506) (D) (529, 576) (E) (576, 552) P y 3 2 1 4 x The CENTRE for EDUCATION in MATHEMATICS and COMPUTING www.cemc.uwaterloo.ca Gauss Contest Grade 7 (The Grade 8 Contest is on the reverse side) Wednesday, May 15, 2013 (in North America and South America) Thursday, May 16, 2013 (outside of North America and South America) Time: 1 hour Calculators are permitted. Instructions ©2012 University of Waterloo 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: 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. 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. The name, school and location of some top-scoring students will be published on the Web site, http://www.cemc.uwaterloo.ca. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. The value of (5 × 3) − 2 is (A) 5 2. 6. 9. (B) 360 (B) 1 (A) 160 (B) 70 (D) 20 (E) 80 (C) 35 (D) 45 (E) 55 (C) 3.6 (D) 0.036 (E) 0.0036 (C) 88 (E) −34 (D) 0 (C) 110 x 20 P Q Nick has six nickels (5¢ coins), two dimes (10¢ coins) and one quarter (25¢ coin). In cents (¢), how much money does Nick have? (C) 35 1 2 1 5 7 The smallest number in the set 2 , 3 , 4 , 6 , 12 (A) 8. (B) 40 If P Q is a straight line segment, then the value of x is (A) 65 7. (E) 13 The value of 1 + 1 − 2 + 3 + 5 − 8 + 13 + 21 − 34 is (A) −32 5. (D) 8 Thirty-six hundredths is equal to (A) 0.36 4. (C) 6 Which of the following numbers is a multiple of 9? (A) 50 3. (B) 9 1 2 (B) 75 (B) 2 3 (C) 1 4 (D) 15 is (D) 5 6 Ahmed is going to the store. One quarter of the way to the store, he stops to talk with Kee. He then continues for 12 km and reaches the store. How many kilometres does he travel altogether? (A) 15 (B) 16 (D) 48 (E) 20 (E) 7 12 Store Start Kee 12 km (C) 24 An expression that produces the values in the second row of the table shown, given the values of n in the first row, is (A) 3n − 2 (B) 2(n − 1) (C) n + 4 (D) 2n (E) 55 (E) 2n − 1 n value 1 1 2 3 3 5 4 7 5 9 10. U V W and XY Z are each 3-digit integers. U, V, W, X, Y, and Z are different digits chosen from the integers 1 to 9. What is the largest possible value for U V W −XY Z? (A) 678 (B) 864 (C) 885 (D) 888 (E) 975 Part B: Each correct answer is worth 6. 11. The length of each edge of a cube is 1 cm. The surface area of the cube, in cm2 , is (A) 24 (B) 1 (C) 4 (D) 12 (E) 6 Grade 7 12. Which of the following pairs of numbers has a greatest common factor of 20? (A) 200 and 2000 (D) 20 and 25 (B) 40 and 50 (E) 40 and 80 (C) 20 and 40 13. Jack, Kelly, Lan, Mihai, and Nate are sitting in the 5 chairs around a circular table. Lan and Mihai are sitting beside each other. Jack and Kelly are not sitting beside each other. The 2 people who are seated on either side of Nate are (A) Jack and Lan (D) Lan and Mihai (B) Jack and Kelly (E) Mihai and Jack (C) Kelly and Mihai 14. If x = 4 and 3x + 2y = 30, what is the value of y? (A) 18 (B) 6 (C) 3 (D) 4 (E) 9 15. Daniel begins with 64 coins in his coin jar. Each time he reaches into the jar, he removes half of the coins that are in the jar. How many times must he reach in and remove coins from his jar so that exactly 1 coin remains in the jar? (A) 5 (B) 32 (C) 6 (D) 7 (E) 63 16. The mean (average) of five consecutive even numbers is 12. The mean of the smallest and largest of these numbers is (A) 12 (B) 10 (C) 14 (D) 8 (E) 16 17. For every 3 chocolates that Claire buys at the regular price, she buys a fourth chocolate for 25 cents. Claire buys 12 chocolates in total for $6.15. What is the regular price of one chocolate, in cents? (A) 180 (B) 45 (C) 60 (D) 54 18. JKLM is a square and P QRS is a rectangle. If JK is parallel to P Q, JK = 8 and P S = 2, then the total area P of the shaded regions is S (A) 32 (B) 16 (C) 56 (D) 48 (E) 57 J K Q R M L (E) 62 19. A special six-sided die is rolled. The probability of rolling a number that is a multiple of three is 12. The probability of rolling an even number is 31. A possibility for the numbers on the die is (A) 1, 2, 3, 5, 5, 6 (D) 1, 2, 3, 3, 4, 6 (B) 1, 2, 3, 3, 5, 6 (E) 2, 3, 3, 3, 5, 6 20. Toothpicks are used to make rectangular grids, as shown. Note that a total of 31 identical toothpicks are used in the 1 × 10 grid. How many toothpicks are used in a 43 × 10 grid? (A) 913 (B) 860 (D) 903 (E) 946 (C) 871 (C) 1, 2, 3, 4, 6, 6 1 x 10 2 x 10 3 x 10 Grade 7 Part C: Each correct answer is worth 8. 7 7P 6 QP + QQP 1PP 7 21. In the addition shown, P and Q each represent single digits, and the sum is 1P P 7. What is P + Q? (A) 9 (B) 12 (D) 15 (E) 13 (C) 14 22. An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, 2, 4, 6, 8 and 1, 4, 7, 10 are arithmetic sequences. In the grid shown, the numbers in each row must form an arithmetic sequence and the numbers in each column must form an arithmetic sequence. The value of x is (A) 37 (B) 28 (D) 43.75 (E) 46 (D) 3 16 5 32 (B) (E) 3 8 7 32 4 25 7 x 10 36 (C) 36 23. In the right-angled triangle P QR, P Q = QR. The segments QS, T U and V W are perpendicular to P R, and the segments ST and U V are perpendicular to QR, as shown. What fraction of 4P QR is shaded? (A) 1 (C) P S U 5 16 W Q T V R 24. One face of a cube contains a circle, as shown. This cube rolls without sliding on a four by four checkerboard. The cube always begins a path on the bottom left square in the position shown and completes the path on the top right square. During each move, an edge of the cube remains in contact with the board. Each move of the cube is either to the right or up. For each path, a face of the cube contacts seven different squares on the checkerboard, including the bottom left and top right squares. The number of different squares that will not be contacted by the face with the circle on any path is (A) 9 (B) 11 (C) 8 (D) 12 (E) 10 up right 25. A box contains a total of 400 tickets that come in five colours: blue, green, red, yellow and orange. The ratio of blue to green to red tickets is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one colour have been selected? (A) 50 (B) 246 (C) 148 (D) 196 (E) 115 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING www.cemc.uwaterloo.ca Gauss Contest Grade 7 (The Grade 8 Contest is on the reverse side) Wednesday, May 16, 2012 (in North America and South America) Thursday, May 17, 2012 (outside of North America and South America) Time: 1 hour ©2011 Centre for Education in Mathematics and Computing Calculators are permitted. 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 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: 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. 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. Please see our Web site: http://www.cemc.uwaterloo.ca. The Gauss Report will list the names of some top-scoring students. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. The value of 202 − 101 + 9 is equal to (A) 120 2. 6. 2 6 1 2 (B) 1 6 (B) 1 3 (A) 60◦ (B) 120◦ (D) 300◦ (E) 180◦ (D) 33 000 000 (E) 330 000 000 (C) 5 6 1 1 1 1 1 2 , 3 , 4 , 5 , 10 (C) 14 (D) 3 6 (E) 4 6 (D) 1 5 (E) 1 10 is (C) 30◦ 120 o 60 o (B) 10 (C) 60 (D) 30 (E) 25 Which of the following statements is true? (B) 7 is less than −1 (E) −8 is less than −2 (C) 10 is less than 1 4 Bailey scores on six of her eight shots. The percentage of shots that she does not score on is (A) 2 9. (E) 92 Fifteen times a number equals three hundred. The number is (A) 0 is less than −5 (D) −1 is less than −3 8. (C) 33 000 Two straight lines intersect as shown. The measure of the angle marked  is (A) 20 7. (B) 330 000 The largest fraction in the set (A) 5. (D) 109 A six-sided die has the numbers one to six on its sides. What is the probability of rolling a five? (A) 4. (C) 111 Which of the following numbers is equal to 33 million? (A) 3 300 000 3. (B) 110 (B) 40 (C) 10 (D) 20 Ben recorded the number of visits to his website from Monday to Friday as shown in the bar graph. The mean (average) number of visits per day to his website over the 5 days is (A) less than 100 (B) between 100 and 200 (C) between 200 and 300 (D) between 300 and 400 (E) more than 400 (E) 25 Visits to Ben s Website Number of Visits 1. 400 300 200 100 M Tu W Th F Days of the Week Grade 7 10. Using the graph, the number of seconds required for a vehicle to travel a total distance of 100 m is (B) 20 (D) 10 (E) 5 Speed (m/s) (A) 2.5 Vehicle s Speed vs. Time Graph (C) 8 30 20 10 1 2 3 4 5 6 Time (s) Part B: Each correct answer is worth 6. 11. The perimeter of a square is 36 cm. The area of the square, in cm2 , is (A) 24 (B) 81 (C) 36 12. Which of the following is not equal to (A) 3.75 (B) 14+1 3+1 (C) 3 4 (D) 1296 (E) 324 15 4 ? +3 (D) 5 4 × 3 4 (E) 21 4 13. On the spinner shown, P Q passes through centre O. If areas labelled R and S are equal, then what percentage of the time will a spin stop on the shaded region? (A) 50% (B) 22.5% (D) 45% (E) 12.5% − 5 4 − 1 4 P R S (C) 25% O Q 14. The digits 2, 4, 6 and 8 are each used once to create two 2-digit numbers. What is the largest possible difference between the two 2-digit numbers? (A) 66 (B) 62 (C) 58 (D) 44 (E) 36 15. If snow falls at a rate of 1 mm every 6 minutes, then how many hours will it take for 1 m of snow to fall? (A) 33 (B) 60 (C) 26 (D) 10 (E) 100 16. The number 503 is a prime number. How many positive integers are factors of 2012? (A) 2 (B) 3 (C) 7 (D) 6 (E) 8 17. The ratio of boys to girls at Gauss Public School is 8 : 5. If there are 128 boys at the school, then how many students are there at the school? (A) 218 (B) 253 (C) 208 18. All four scales shown are balanced. One possible replacement for the ? is (A) (B) (D) (E) (C) (D) 133 (E) 198 Grade 7 19. A set of five different positive integers has a mean (average) of 20 and a median of 18. What is the greatest possible integer in the set? (A) 60 (B) 26 (C) 46 (D) 12 (E) 61 20. Chris lies on Fridays, Saturdays and Sundays, but he tells the truth on all other days. Mark lies on Tuesdays, Wednesdays and Thursdays, but he tells the truth on all other days. On what day of the week would they both say: “Tomorrow, I will lie.”? (A) Monday (B) Thursday (C) Friday (D) Sunday (E) Tuesday Part C: Each correct answer is worth 8. 3 21. A triangular prism has a volume of 120 cm3. Two edges of the triangular faces measure 3 cm and 4 cm, as shown. The height of the prism, in cm, is (A) 12 (B) 20 (D) 16 (E) 8 4 (C) 10 22. A quiz has three questions, with each question worth one mark. If 20% of the students got 0 questions correct, 5% got 1 question correct, 40% got 2 questions correct, and 35% got all 3 questions correct, then the overall class mean (average) mark was (A) 1.8 (B) 1.9 (C) 2 (D) 2.1 (E) 2.35 23. The number N is the product of all positive odd integers from 1 to 99 that do not end in the digit 5. That is, N = 1 × 3 × 7 × 9 × 11 × 13 × 17 × 19 × · · · × 91 × 93 × 97 × 99. The units digit of N is (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 24. P QRS is a parallelogram with area 40. If T and V are the midpoints of sides P S and RS respectively, then the area of P RV T is (A) 10 (B) 12 (D) 16 (E) 18 Q R (C) 15 V P T S 25. The positive integers are arranged in rows and columns as shown below. Row Row Row Row Row Row 1 2 3 4 5 6 1 2 4 7 11 16 3 5 8 12 17 6 9 13 18... 10 14 19 15 20 21 More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000? (A) 15 (B) 19 (C) 17 (D) 16 (E) 18 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING www.cemc.uwaterloo.ca Gauss Contest (Grade 7) (The Grade 8 Contest is on the reverse side) Wednesday, May 11, 2011 Time: 1 hour ©2010 Centre for Education in Mathematics and Computing Calculators are permitted. 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 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: 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. 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. Please see our Web site: http://www.cemc.uwaterloo.ca. The Gauss Report will list the names of some top-scoring students. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 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. 2. 3. The value of 5 + 4 − 3 + 2 − 1 is (A) 0 (B) −5 (C) 3 √ The value of 9 + 16 is (A) 5.2 (B) 7 (C) 5.7 (D) −3 (E) 7 (D) 25 (E) 5 (A) 50 (B) 10 (D) 250 (E) 5 (C) 25 Favourite Season 5 4 3 2 1 0 Fall Summer Spring Winter Number of Students Students were surveyed about their favourite season. The results are shown in the bar graph. What percentage of the 10 students surveyed chose Spring? 4. Ground beef sells for $5.00 per kg. How much does 12 kg of ground beef cost? (A) $5.00 (B) $12.00 (C) $60.00 (D) $17.00 (E) $2.40 5. The smallest number in the list {1.0101, 1.0011, 1.0110, 1.1001, 1.1100} is (A) 1.0101 (B) 1.0011 (C) 1.0110 (D) 1.1001 (E) 1.1100 6. You are writing a multiple choice test and on one question you guess and pick an answer at random. If there are five possible choices (A,B,C,D,E), what is the probability that you guessed correctly? (B) 55 (C) 45 (D) 25 (E) 35 (A) 51 7. 1 3 + (A) 1 3 + 1 3 + 3 31 1 3 + 1 3 + 1 3 + (B) 7 + 1 3 1 3 equals (C) 3 7 (D) 7 + 3 (E) 7 × 1 3 8. Keegan paddled the first 12 km of his 36 km kayak trip before lunch. What fraction of his overall trip remains to be completed after lunch? (A) 12 (B) 65 (C) 34 (D) 23 (E) 35 9. If the point (3, 4) is reflected in the x-axis, what are the coordinates of its image? (A) (−4, 3) (B) (−3, 4) (D) (3, −4) (E) (−3, −4) y (3, 4) (C) (4, 3) x Grade 7 10. I bought a new plant for my garden. Anika said it was a red rose, Bill said it was a purple daisy, and Cathy said it was a red dahlia. Each person was correct in stating either the colour or the type of plant. What was the plant that I bought? (A) purple dahlia (D) yellow rose (B) purple rose (E) red daisy (C) red dahlia Part B: Each correct answer is worth 6. 11. In the diagram, the value of x is (A) 15 (B) 20 (D) 18 (E) 36 (C) 22 2x 3x 12. A square has a perimeter of 28 cm. The area of the square, in cm2 , is (A) 196 (B) 784 (C) 64 (D) 49 (E) 56 13. Five children had dinner. Chris ate more than Max. Brandon ate less than Kayla. Kayla ate less than Max but more than Tanya. Which child ate the second most? (A) Brandon (B) Chris (C) Kayla (D) Max (E) Tanya 14. A palindrome is a positive integer that is the same when read forwards or backwards. For example, 545 and 1331 are both palindromes. The difference between the smallest three-digit palindrome and the largest three-digit palindrome is (A) 909 (B) 898 (C) 888 (D) 979 (E) 878 15. A ski lift carries a skier at a rate of 12 km per hour. How many kilometres does the ski lift carry the skier in 10 minutes? (A) 120 (B) 1.2 (C) 2 (D) 2.4 (E) 1.67 16. A 51 cm rod is built from 5 cm rods and 2 cm rods. All of the 5 cm rods must come first, and are followed by the 2 cm rods. For example, the rod could be made from seven 5 cm rods followed by eight 2 cm rods. How many ways are there to build the 51 cm rod? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 17. In Braydon’s cafeteria, the meats available are beef and chicken. The fruits available are apple, pear and banana. Braydon is randomly given a lunch with one meat and one fruit. What is the probability that the lunch will include a banana? (A) 1 3 (B) 2 3 (C) 1 2 (D) 1 5 (E) 3 5 18. Three pumpkins are weighed two at a time in all possible ways. The weights of the pairs of pumpkins are 12 kg, 13 kg and 15 kg. How much does the lightest pumpkin weigh? (A) 4 kg (B) 5 kg (C) 6 kg (D) 7 kg (E) 8 kg Grade 7 19. The sum of four numbers is T. Suppose that each of the four numbers is now increased by 1. These four new numbers are added together and then the sum is tripled. What is the value of this final result? (A) 3T + 3 (B) 3T + 4 (C) 3T + 12 (D) T + 12 (E) 12T 20. A triangular prism is placed on a rectangular prism, as shown. The volume of the combined structure, in cm3 , is (A) 76 (B) 78 (D) 84 (E) 66 (C) 72 3 cm 2 cm 5 cm 4 cm 6 cm Part C: Each correct answer is worth 8. 21. Steve begins at 7 and counts forward by 3, obtaining the list 7, 10, 13, and so on. Dave begins at 2011 and counts backwards by 5, obtaining the list 2011, 2006, 2001, and so on. Which of the following numbers appear in each of their lists? (A) 1009 (B) 1006 (C) 1003 (D) 1001 (E) 1011 22. A pool has a volume of 4000 L. Sheila starts filling the empty pool with water at a rate of 20 L/min. The pool springs a leak after 20 minutes and water leaks out at 2 L/min. Beginning from the time when Sheila starts filling the empty pool, how long does it take until the pool is completely full? (A) 3 hours (D) 4 hours 20 minutes (B) 3 hours 40 minutes (E) 3 hours 20 minutes (C) 4 hours 23. In the addition of the three-digit numbers shown, the letters A, B, C, D, and E each represent a single digit. A B E A C E + A D E 2 0 1 1 The value of A + B + C + D + E is (A) 34 (B) 21 (C) 32 (D) 27 (E) 24 24. From the figure shown, three of the nine squares are to be selected. Each of the three selected squares must share a side with at least one of the other two selected squares. In how many ways can this be done? (A) 19 (B) 22 (D) 16 (E) 20 (C) 15 25. Ten circles are all the same size. Each pair of these circles overlap but no circle is exactly on top of another circle. What is the greatest possible total number of intersection points of these ten circles? (A) 40 (B) 70 (C) 80 (D) 90 (E) 110 Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Gauss Contest (Grade 7) (The Grade 8 Contest is on the reverse side) Wednesday, May 12, 2010 Time: 1 hour ©2009 Centre for Education in Mathematics and Computing Calculators are permitted. 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 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: 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. 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. Please see our Web site: http://www.cemc.uwaterloo.ca. The Gauss Report will list the names of some top-scoring students. You will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation. Grade 7 Scoring: There is no penalty for an incorrect answer. Each unanswered question is worth 2, to a maximum of 10 unanswered questions. 1. (A) 10 (B) 20 (D) 40 (E) 50 Tanya scored 20 out of 25 on her math quiz. What percent did she score? (A) 75 3. (B) 95 (C) 80 (D) 20 (E) 45 (C) 100 (D) 18 (E) 40 The value of 4 × 5 + 5 × 4 is (A) 160 4. (C) 30 Favourite Pet 60 50 40 30 20 10 0 cat bird rabbit dog fish 2. The grade 7 students at Gauss Public School were asked, “What is your favourite pet?” The number of students who chose fish is Number of Students Part A: Each correct answer is worth 5. (B) 400 In the diagram, the point with coordinates (−2, −3) is located at (A) A (B) B (D) D (E) E E B x C D 5. Chaz gets on the elevator on the eleventh floor. The elevator goes down two floors, then stops. Then the elevator goes down four more floors and Chaz gets off the elevator. On what floor does Chaz get off the elevator? (A) 7th floor 6. 8. (B) 9th floor (C) 4th floor (D) 5th floor (E) 6th floor If 10.0003 ×  = 10000.3, the number that should replace the  is (A) 100 7. A (C) C (B) 1000 (C) 10000 (D) 0.001 (E) 0.0001 In the diagram, the value of x is (A) 40 (B) 35 (D) 30 (E) 25 150 (C) 150 x How many 1 cm × 1 cm × 1 cm blocks are needed to build the solid rectangular prism shown? (A) 10 (B) 12 (D) 66 (E) 36 3 cm (C) 33 3 cm 4 cm Grade 7 9. The time on a digital clock reads 3:33. What is the shortest length of time, in minutes, until all of the digits are again equal to each other? (A) 71 (B) 60 (C) 142 (D) 222 (E) 111 10. Each number below the top row is the product of the number to the right and the number to the left in the row immediately above it. What is the value of x? (A) 8 (B) 4 (D) 5 (E) 6 7 700 2 Part B: Each correct answer is worth 6. 11. The area of the figure, in square units, is (A) 36 (B) 64 (D) 58 (E) 32 y 35 (C) 7 x 5 3 6 (C) 46 5 12. Recycling 1 tonne of paper will save 24 trees. If 4 schools each recycle of paper, then the total number of trees this will save is (A) 24 (B) 72 (C) 18 (D) 126 3 4 of a tonne (E) 80 13. If the mean (average) of five consecutive integers is 21, the smallest of the five integers is (A) 17 (B) 21 (C) 1 (D) 18 (E) 19 14. A bag