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2017 Gauss Contest
Grade 7

Wednesday, May 10, 2017
(in North America and South America)

Thursday, May 11, 2017
(outside of North American and South America)

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©2016 University of Waterloo

Instructions

Time: 1 hour

Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.

  1. Do not open the Contest booklet until you are told to do so.
  2. You may use rulers, compasses and paper for rough work.
  3. Be sure that you understand the coding system for your answer sheet. If you are not sure, ask your teacher to explain it.
  4. This is a multiple-choice test. Each question is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. When you have made your choice, enter the appropriate letter for that question on your answer sheet.
    5. Scoring:
    1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
    2. There is no penalty for an incorrect answer.
    3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
  5. Diagrams are not drawn to scale. They are intended as aids only.
  6. 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.
Scoring:
  1. There is no penalty for an incorrect answer.
  2. 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 \((2+4+6) - (1+3+5)\) is

    1. \(0\)
    2. \(3\)
    3. \(-3\)
    4. \(21\)
    5. \(111\)

  2. Based on the graph shown, which sport is played by the most students?

    A bar graph entitled Sports Played by Students. There are five bars corresponding to five sports with the following numbers of students: Hockey at 40, Basketball at 30, Soccer at 50, Volleyball at 20, and Badminton at 10.

    1. hockey
    2. basketball
    3. soccer
    4. volleyball
    5. badminton

  3. Michael has $280 in $20 bills. How many $20 bills does he have?

    1. \(10\)
    2. \(12\)
    3. \(14\)
    4. \(16\)
    5. \(18\)

  4. When two integers between 1 and 10 are multiplied, the result is 14. What is the sum of these two integers?

    1. \(2\)
    2. \(5\)
    3. \(7\)
    4. \(9\)
    5. \(33\)

  5. Three thousandths is equal to

    1. \(300\)
    2. \(0.3\)
    3. \(0.03\)
    4. \(30\)
    5. \(0.003\)

  6. Consider the following diagram.

    A square with four equal sides marked. One diagonal is drawn. The angle between the diagonal and one of the sides is marked x degrees.

    In the square shown, \(x\) is equal to

    1. \(0\)
    2. \(45\)
    3. \(60\)
    4. \(180\)
    5. \(360\)

  7. Which integer is closest in value to \(\frac{35}{4}\)?

    1. \(10\)
    2. \(8\)
    3. \(9\)
    4. \(7\)
    5. \(6\)

  8. When \(n=101\), which of the following expressions has an even value?

    1. \(3n\)
    2. \(n+2\)
    3. \(n-12\)
    4. \(2n-2\)
    5. \(3n+2\)

  9. The sum of three consecutive integers is 153. The largest of these three integers is

    1. \(52\)
    2. \(50\)
    3. \(53\)
    4. \(54\)
    5. \(51\)

  10. In the diagram, \(\triangle PQR\) is equilateral and is made up of four smaller equilateral triangles.

    One of the smaller equilateral triangles is in the centre. The remaining three smaller equilateral triangles are arranged so that each shares a side with the centre triangle.

    If each of the smaller triangles has a perimeter of 9 cm, what is the perimeter of \(\triangle PQR\)?

    1. \(15 \text{ cm}\)
    2. \(9\text{ cm}\)
    3. \(36\text{ cm}\)
    4. \(27\text{ cm}\)
    5. \(18\text{ cm}\)

Part B: Each correct answer is worth 6.

  1. The number that goes into the \(\Box\) to make \(\dfrac{3}{7} = \dfrac{\Box}{63}\) true is

    1. \(27\)
    2. \(9\)
    3. \(59\)
    4. \(63\)
    5. \(3\)

  2. 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?

    1. \(\$210\)
    2. \(\$230\)
    3. \(\$250\)
    4. \(\$220\)
    5. \(\$200\)

  3. When the shaded triangle shown is translated, which of the following triangles can be obtained?

    A coordinate plane with six right-angled triangles of the same shape and size but different orientations. One triangle is shaded and the remaining five are unshaded and labelled A, B, C, D and E. A description of the triangles follows.

    1. \(A\)
    2. \(B\)
    3. \(C\)
    4. \(D\)
    5. \(E\)

  4. 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)? 

    1. 7:20 p.m.
    2. 5:00 p.m.
    3. 6:20 p.m.
    4. 5:20 p.m.
    5. 8:50 p.m.

  5. 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?

    1. Faiz
    2. Henry
    3. Omar
    4. Ryan
    5. Toma

  6. A circular spinner is divided into 20 equal sections, as shown.

    Each section is labelled with a number from 1 to 20.

    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?

    1. \(\frac{12}{20}\)
    2. \(\frac{14}{20}\)
    3. \(\frac{15}{20}\)
    4. \(\frac{7}{20}\)
    5. \(\frac{6}{20}\)

  7. The mean (average) of the four integers \(78,83,82,\) and \(x\) is 80. Which one of the following statements is true?

    1. \(x\) is 2 greater than the mean
    2. \(x\) is 1 less than the mean
    3. \(x\) is 2 less than the mean
    4. \(x\) is 3 less than the mean
    5. \(x\) is equal to the mean

  8. 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?

    1. A discount of 20%
    2. A discount of 10%, then a discount of 10% off the new price
    3. A discount of 15%, then a discount of 5% off the new price
    4. A discount of 5%, then a discount of 15% off the new price
    5. All four options above give the same price

  9. Two sheets of \(11\text{ cm}\times 8\text{ cm}\) paper are placed on top of each other, forming an overlapping \(8\text{ cm}\times 8\text{ cm}\) square in the centre, as shown.

    Rectangle XYZW with longer sides WX and ZY horizontal and shorter sides WZ and XY vertical. A solid vertical line is drawn from WX to ZY near the left end of the rectangle. A dashed vertical line is drawn from WX to ZY near the right end of the rectangle. These lines divide the rectangle into three regions: a square in the centre with two rectangular strips on either end.

    The area of rectangle \(WXYZ\) is

    1. \(88\,\text{cm}^2\)
    2. \(112\,\text{cm}^2\)
    3. \(136\,\text{cm}^2\)
    4. \(121\,\text{cm}^2\)
    5. \(176\,\text{cm}^2\)

  10. Betty and Ann are walking around a rectangular park with dimensions 600 m by 400 m, as shown.

    The top and bottom edges of the rectangle are 600 metres. The left and right edges are 400 metres. The top left corner of the rectangle is labelled Start. An arrow labelled Betty points right and an arrow labelled Ann points down.

    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?

    1. \(5:3\)
    2. \(9:4\)
    3. \(11:6\)
    4. \(12:5\)
    5. \(17:7\)

Part C: Each correct answer is worth 8.

  1. Rectangles that measure \(4\times 2\) 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.

    The top and bottom edges of each rectangle measure 4 units while the left and right edges measure 2 units.

    When a total of ten rectangles are arranged in this pattern, what is the perimeter of the figure?

    1. \(48\)
    2. \(64\)
    3. \(90\)
    4. \(84\)
    5. \(100\)

  2. In the six-digit number \(1ABCDE\), each letter represents a digit. Given that \(1ABCDE \times 3 = ABCDE1\), the value of \(A+B+C+D+E\) is

    1. \(29\)
    2. \(26\)
    3. \(22\)
    4. \(30\)
    5. \(28\)

  3. 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?

    1. \(27\)
    2. \(29\)
    3. \(35\)
    4. \(26\)
    5. \(28\)

  4. 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

    1. \(60\)
    2. \(63\)
    3. \(67\)
    4. \(70\)
    5. \(72\)

  5. 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?

    1. \(542\,708\)
    2. \(543\,213\)
    3. \(542\,203\)
    4. \(543\,326\)
    5. \(543\,618\)