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2019 Gauss Contest
Grade 8

Wednesday, May 15, 2019
(in North America and South America)

Thursday, May 16, 2019
(outside of North American and South America)

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©2018 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. Ali ate half of a muffin. What percentage of the muffin did Ali eat?

    1. \(10\%\)
    2. \(17\%\)
    3. \(21\%\)
    4. \(40\%\)
    5. \(50\%\)

  2. In the triangle shown, the value of \(x\) is

    A triangle with three equal sides and three angles, each marked x degrees.

    1. \(30\)
    2. \(60\)
    3. \(45\)
    4. \(90\)
    5. \(55\)

  3. Which of the following integers is closest to 0?

    1. \(-1\)
    2. \(2\)
    3. \(-3\)
    4. \(4\)
    5. \(-5\)

  4. Which of these numbers gives a remainder of 3 when divided by 5?

    1. \(51\)
    2. \(64\)
    3. \(76\)
    4. \(88\)
    5. \(99\)

  5. How many integers between 10 and 20 are prime numbers?

    1. \(0\)
    2. \(1\)
    3. \(2\)
    4. \(3\)
    5. \(4\)

  6. Based on the graph shown, how many vehicles had an average speed of at least 80 km/h?

    A bar graph titled Average Vehicle SPeed. It shows Speed in kilometres per hour and the Number of Vehicles. The number of vehicles at each speed is outlined in the following description.

    1. \(45\)
    2. \(15\)
    3. \(35\)
    4. \(70\)
    5. \(50\)

  7. How many positive integers less than 100 are divisible by both 3 and 7?

    1. \(2\)
    2. \(3\)
    3. \(4\)
    4. \(5\)
    5. \(6\)

  8. The circumference of a circle is 100. The diameter of this circle is equal to

    1. \(100\times \pi\)
    2. \(\dfrac{2\pi}{ 100}\)
    3. \(\dfrac{100}{ \pi}\)
    4. \(2\pi \times 100\)
    5. \(\dfrac{\pi}{100}\)

  9. In the diagram, point \(F\) has coordinates \((6,6)\).

    A coordinate plane with x and y axes. Eight points labelled A, B, C, D, E, F, P, and Q are plotted. P is 1 unit right from the origin and Q is 5 units right from the origin. A line segment joins P and Q. The rest of the points are described as follows:

    Points \(P\) and \(Q\) are two vertices of a triangle. Which of the following points can be joined to \(P\) and \(Q\) to create a triangle with an area of 6?

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

  10. 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?

    1. \(\$3.75\)
    2. \(\$3.90\)
    3. \(\$3.70\)
    4. \(\$3.40\)
    5. \(\$3.95\)

Part B: Each correct answer is worth 6.

  1. Two of the side lengths in an isosceles triangle are 6 and 8. The perimeter of the triangle could be

    1. \(18\)
    2. \(14\)
    3. \(22\)
    4. \(16\)
    5. \(24\)

  2. Line segments \(PQ\) and \(RS\) intersect as shown. What is the value of \(x+y\)?

    The two angles that lie along the line segment PQ measure 60 degrees and y plus 5 degrees. The two angles that lie along the line segment RS measure 60 degrees and 4x degrees.

    1. \(145\)
    2. \(70\)
    3. \(130\)
    4. \(85\)
    5. \(240\)

  3. The mean (average), the median and the mode of the five numbers \(12,9,11,16,x\) are all equal. What is the value of \(x\)?

    1. \(9\)
    2. \(11\)
    3. \(12\)
    4. \(13\)
    5. \(16\)

  4. The two equal-arm scales shown are balanced.

    The first scale has three circles on the left side and two triangles on the right side. The second scale has one square, one circle, and one triangle on the left side. On the right side are two squares.

    Of the following, one circle and three triangles has the same mass as

    1. Three squares
    2. Two triangles and two squares
    3. Four circles
    4. Three circles and one squares
    5. Three circles

  5. A spinner is divided into 3 equal sections, as shown.

    The sections are blue, green, and red.

    An arrow is attached to the centre of the spinner. The arrow is spun twice. What is the probability that the arrow lands on the same colour twice?

    1. \(\dfrac{1}{9}\)
    2. \(\dfrac{2}{3}\)
    3. \(\dfrac{1}{2}\)
    4. \(\dfrac{1}{3}\)
    5. \(\dfrac{2}{9}\)

  6. A Gauss brand light bulb will work for \(24\,999\) hours. If it is used for exactly 2 hours every day starting on a Monday, on what day of the week will it stop working?

    1. \(\text{Thursday}\)
    2. \(\text{Friday}\)
    3. \(\text{Saturday}\)
    4. \(\text{Sunday}\)
    5. \(\text{Monday}\)

  7. Each of \(w,x,y\), and \(z\) is an integer. If \(w+x=45\), \(x+y=51\), and \(y+z=28\), what is the value of \(w+z\)?

    1. \(28\)
    2. \(22\)
    3. \(17\)
    4. \(23\)
    5. \(15\)

  8. 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?

    1. \(\text{Bruce owns the fewest cats.}\)
    2. \(\text{Bruce owns the most cats.}\)
    3. \(\text{Kathy owns the most cats.}\)
    4. \(\text{Alice owns the most dogs.}\)
    5. \(\text{Kathy owns the fewest dogs.}\)

  9. A line segment joins the points \(P(-4,1)\) and \(Q(1,-11)\). What is the length of \(PQ\)?

    1. \(13\)
    2. \(12\)
    3. \(12.5\)
    4. \(13.6\)
    5. \(12.6\)
  10. \(PQRS\) is a square with side length 60 and centre \(C\). Point \(W\) lies on \(PS\) so that \(WS=53\). Point \(X\) lies on \(SR\) so that \(XR=40\). The midpoint of \(QR\) is \(Y\). Point \(Z\) lies on \(PQ\).

    Four lines are drawn connecting centre C to each of the points Z, Q, W, and X. This divides the interior of the square PQRS into four quadrilateral regions. Quadrilaterals CYQZ and CXSW are shaded. Quadrilaterals CYRX and CZPQ are not shaded.

    What is the length of \(ZQ\) so that the total area of the shaded regions is equal to the total area of the non-shaded regions?

    1. \(21\)
    2. \(15\)
    3. \(23\)
    4. \(19\)
    5. \(17\)

Part C: Each correct answer is worth 8.

  1. In Jen’s baseball league, each team plays exactly 6 games against each of the other teams in the league. If a total of 396 games are played, how many teams are in the league?

    1. \(12\)
    2. \(16\)
    3. \(15\)
    4. \(13\)
    5. \(9\)

  2. Rich chooses a 4-digit positive integer. He erases one of the digits of this integer. The remaining digits, in their original order, form a 3-digit positive integer. When Rich adds this 3-digit integer to the original 4-digit integer, the result is 6031. What is the sum of the digits of the original 4-digit integer?

    1. \(18\)
    2. \(20\)
    3. \(22\)
    4. \(19\)
    5. \(21\)

  3. If \(n\) is a positive integer, the notation \(n!\) (read “\(n\) factorial”) is used to represent the product of the integers from 1 to \(n\) inclusive. For example, \(5!=1\times 2\times 3\times 4\times 5=120\). Which of the following is equal to a perfect square?

    1. \(\dfrac{(20!)(19!)}{1}\)
    2. \(\dfrac{(20!)(19!)}{2}\)
    3. \(\dfrac{(20!)(19!)}{3}\)
    4. \(\dfrac{(20!)(19!)}{4}\)
    5. \(\dfrac{(20!)(19!)}{5}\)

  4. There are many ways in which the list \(0,1,2,3,4,5,6,7,8,9\) can be separated into groups. For example, this list could be separated into the four groups \(0,3,4,8\) and \(1,2,7\) and \(6\) and \(5,9\). The sum of the numbers in each of these four groups is \(15, 10, 6\), and 14, respectively. In how many ways can the list \(0,1,2,3,4,5,6,7,8,9\) be separated into at least two groups so that the sum of the numbers in each group is the same?

    1. \(26\)
    2. \(29\)
    3. \(24\)
    4. \(27\)
    5. \(32\)

  5. In quadrilateral \(PQRS\), diagonals \(PR\) and \(SQ\) intersect at \(O\) inside \(PQRS\), \(SP=SQ=SR=1\), and \(\angle QSR=2\angle QSP\).

    Marc determines the measure of the twelve angles that are the interior angles of \(\triangle POS\), \(\triangle POQ\), \(\triangle ROS\), and \(\triangle ROQ\). The measure of each of these angles, in degrees, is a positive integer, and exactly six of these integers are prime numbers. How many different quadrilaterals have these properties and are not rotations or translations of each other?

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