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2016 Gauss Contest
Grade 8

Wednesday, May 11, 2016
(in North America and South America)

Thursday, May 12, 2016
(outside of North American and South America)

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©2015 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 \(444-44-4\) is

    1. \(396\)
    2. \(402\)
    3. \(392\)
    4. \(400\)
    5. \(408\)

  2. Which of the following is equal to \(\frac{4}{5}\)?

    1. \(4.5\)
    2. \(0.8\)
    3. \(80.0\)
    4. \(0.08\)
    5. \(0.45\)

  3. The graph shows the number of hours Stan worked on a school project.

    A bar graph titled Stan's Time Working on a Project. it shows the days of the week on the horizontal axis and hours worked on the vertical axis. The number of hours spent on each day is given in the following list.

    For how many hours in total did he work on the project?

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

  4. Three tenths plus four thousandths is equal to

    1. \(4030\)
    2. \(0.0403\)
    3. \(0.304\)
    4. \(0.34\)
    5. \(30.004\)

  5. A cube is created by folding the figure shown.

    There are six identical squares in the net. Each square has one integer in it. The middle of the net is formed by four squares arranged in a column. From top to bottom, the integers on those squares are 1, 2, 3, and 4. There is one square on either side of the third square down. The integer 5 is on the square to the left and the integer 6 is on the square to the right.

    Which face is opposite the face with 1 on it?

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

  6. In \(\triangle PQR\) shown, side \(PR\) is horizontal and side \(PQ\) is vertical.

    The coordinates of Q are (negative 11, negative 8) and the coordinates of R are (negative 6, negative 2).

    The coordinates of \(P\) are

    1. \((-8,-2)\)
    2. \((-6,-8)\)
    3. \((-11,-6)\)
    4. \((-11,-2)\)
    5. \((-8,-6)\)

  7. A rectangle with a width of 2 cm and a length of 18 cm has the same area as a square with a side length of

    1. \(6 \textrm{ cm}\)
    2. \(12 \textrm{ cm}\)
    3. \(9 \textrm{ cm}\)
    4. \(10 \textrm{ cm}\)
    5. \(8 \textrm{ cm}\)

  8. Gaby lists the numbers 3, 4, 5, 6, 7, 8, and 9. In her list, the ratio of the number of prime numbers to the number of composite numbers is

    1. \(3:4\)
    2. \(5:2\)
    3. \(2:5\)
    4. \(3:7\)
    5. \(1:6\)

  9. 10% of 200 is equal to 20% of

    1. \(40\)
    2. \(50\)
    3. \(100\)
    4. \(400\)
    5. \(800\)

  10. The circumference of a circle is \(100\pi\) cm. What is the radius of the circle?

    1. \(20\,\textrm{cm}\)
    2. \(100\,\textrm{cm}\)
    3. \(50\,\textrm{cm}\)
    4. \(25\,\textrm{cm}\)
    5. \(10\,\textrm{cm}\)

Part B: Each correct answer is worth 6.

  1. In the diagram, \(\triangle PQR\) is right-angled.

    Angle PQR equals 90 degrees.

    Point \(S\) lies on \(PR\) so that \(\triangle QRS\) is equilateral and \(\triangle PQS\) is isosceles with \(PS=QS\). The measure of \(\angle QPR\) is

    1. \(35\degree\)
    2. \(37.5\degree\)
    3. \(25\degree\)
    4. \(32.5\degree\)
    5. \(30\degree\)

  2. Operations are placed in each \(\bigcirc\) so that \(3 \bigcirc 5 \bigcirc 7 \bigcirc 9 = 78\). Listed from left to right, the operations are

    1. \(+, \times, +\)
    2. \(+, +, \times\)
    3. \(\times, \times, -\)
    4. \(\times, \times, +\)
    5. \(\times, +, \times\)

  3. 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?

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

  4. One soccer ball and one soccer shirt together cost $100. Two soccer balls and three soccer shirts together cost $262. What is the cost of one soccer ball?

    1. \(\$38\)
    2. \(\$50\)
    3. \(\$87.30\)
    4. \(\$45\)
    5. \(\$40\)

  5. A map has a scale of \(1:600\,000\). On the map, the distance between Gausstown and Piville is 2 cm. What is the actual distance between the towns?

    1. \(12\,\textrm{km}\)
    2. \(1.2\,\textrm{km}\)
    3. \(120\,\textrm{km}\)
    4. \(1200\,\textrm{km}\)
    5. \(12\,000\,\textrm{km}\)

  6. The mean (average) of a set of six numbers is 10. If the number 25 is removed from the set, the mean of the remaining numbers is

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

  7. How many positive integers between 10 and 2016 are divisible by 3 and have all of their digits the same?

    1. \(9\)
    2. \(12\)
    3. \(6\)
    4. \(18\)
    5. \(3\)

  8. Joe filled up his car’s gas tank. After travelling 165 km, \(\frac{3}{8}\) of the gas in the tank was used. At this rate, approximately how much farther can the car travel before its fuel tank is completely empty?

    1. \(99\, \textrm{km}\)
    2. \(440\,\textrm{km}\)
    3. \(605\,\textrm{km}\)
    4. \(264\, \textrm{km}\)
    5. \(275\, \textrm{km}\)

  9. The two scales shown are balanced.

    In the first scale, two circles are on the left and six squares are on the right. In the second scale, two circles and six squares on are on th left and four triangles are on the right.

    Which of the following is not true?

    1. One circle equals one triangle
    2. Two triangles equal a circle and three squares
    3. One circle equals three squares
    4. A circle and a triangle equal four squares
    5. A triangle equals three squares

  10. In the diagram, what is the length of \(BD\)?

    The coordinate grid with the vertices of quadrilateral ABCD plotted. The coordinates of point A are (negative 2, 9), of point B are (3, 9), of point C are (3, negative 3) and of point D are (negative 2, negative 3).

    1. \(13\)
    2. \(17\)
    3. \(\sqrt{205}\)
    4. \(\sqrt{160}\)
    5. \(15\)

Part C: Each correct answer is worth 8.

  1. Two 5-digit positive integers are formed using each of the digits from 0 through 9 once. What is the smallest possible positive difference between the two integers?

    1. \(469\)
    2. \(269\)
    3. \(247\)
    4. \(229\)
    5. \(249\)

  2. In rectangle \(ABCD\), what is the total area of the shaded region?

    Starting at point A and moving around the rectangle, the vertices are labelled A, B, C, and D, in order. AD equals 6 centimeters. Point F is on BC such that BF equals 5 centimeters and point H is on CD such that CH equals 6 centimeters and HD equals 4 centimeters. Point E is inside rectangle ABCD such that EH equals 2 centimeters and angle EHD equals 90 degrees. Point G is on AB and AE equals EG. Angle AEG measures 90 degrees. Quadrilateral ADHE and quadrilateral BGEF are both shaded.

    1. \(25 \textrm{cm}^2\)
    2. \(31 \textrm{cm}^2\)
    3. \(39 \textrm{cm}^2\)
    4. \(35 \textrm{cm}^2\)
    5. \(41 \textrm{cm}^2\)

  3. Zeus starts at the origin \((0,0)\) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. For example, he cannot move from \((0,0)\) to \((1,0)\) to \((2,0)\). What is the smallest number of moves that he can make to get to the point \((1056,1007)\)?

    1. \(2112\)
    2. \(2161\)
    3. \(2063\)
    4. \(2111\)
    5. \(2113\)

  4. What is the tens digit of \(3^{2016}\)?

    1. \(0\)
    2. \(2\)
    3. \(4\)
    4. \(6\)
    5. \(8\)

  5. In the table, the numbers in each row form an arithmetic sequence when read from left to right.

    18
    43
    40
    \(x\) 26

    Similarly, the numbers in each column form an arithmetic sequence when read from top to bottom. What is the sum of the digits of the value of \(x\)?

    (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, \(3,5,7,9\) are the first four terms of an arithmetic sequence.)

    1. \(5\)
    2. \(2\)
    3. \(10\)
    4. \(7\)
    5. \(13\)