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2020 Pascal Contest
(Grade 9)

Tuesday, February 25, 2020
(in North America and South America)

Wednesday, February 26, 2020
(outside of North American and South America)

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©2019 University of Waterloo

Instructions

Time: 60 minutes

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

  1. Do not open the Contest booklet until you are told to do so.
  2. You may use rulers, compasses and paper for rough work.
  3. Be sure that you understand the coding system for your response form. If you are not sure, ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in circles completely.
  4. On your response form, print your school name and city/town in the box in the upper right corner.
  5. Be certain that you code your name, age, grade, and the Contest you are writing in the response form. Only those who do so can be counted as eligible students.
  6. 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. After making your choice, fill in the appropriate circle on the response form.
  7. 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.
  8. Diagrams are not drawn to scale. They are intended as aids only.
  9. When your supervisor tells you to begin, you will have sixty minutes of working time.
  10. You may not write more than one of the Pascal, Cayley and Fermat Contests in any given year.
Do not discuss the problems or solutions from this contest online for the next 48 hours.
The name, grade, school and location, and score range of some top-scoring students will be published on the website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some students may be shared with other mathematical organizations for other recognition opportunities.
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. How many ☐ symbols are in the figure?

    There are five groups, each with five squares.

    1. \(24\)
    2. \(20\)
    3. \(15\)
    4. \(17\)
    5. \(25\)

  2. The value of \(0.8+0.02\) is

    1. \(0.28\)
    2. \(8.02\)
    3. \(0.82\)
    4. \(0.16\)
    5. \(0.01\)

  3. If \(2x + 6 = 16\), the value of \(x+4\) is

    1. \(7\)
    2. \(8\)
    3. \(9\)
    4. \(15\)
    5. \(13\)

  4. When two positive integers are multiplied, the result is 24. When these two integers are added, the result is 11. When the smaller integer is subtracted from the larger integer, the result is

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

  5. In the diagram, \(\triangle PQR\) has side lengths as shown.

    In triangle PQR, PQ is labelled x-1, QR is labelled x+1, and PR is labelled 7.

    If \(x = 10\), the perimeter of \(\triangle PQR\) is

    1. \(29\)
    2. \(31\)
    3. \(25\)
    4. \(27\)
    5. \(23\)

  6. The value of \(\dfrac{2^4-2}{2^3-1}\) is

    1. \(1\)
    2. \(0\)
    3. \(\frac{7}{4}\)
    4. \(\frac{4}{3}\)
    5. \(2\)

  7. Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is \(3, 14, 25, 36, \ldots\). A number that will appear in Ewan’s sequence is

    1. \(113\)
    2. \(111\)
    3. \(112\)
    4. \(110\)
    5. \(114\)

  8. Matilda counted the birds that visited her bird feeder yesterday. She summarized the data in the bar graph shown.

    The vertical axis of a bar graph is labelled 'Number' and the horizontal axis labelled 'Type of Bird'. There are three bars on the graph: Goldfinch has a height of 6, Sparrow has a height of 9 and Grackle has a height of 5.

    The percentage of birds that were goldfinches is

    1. \(15\%\)
    2. \(20\%\)
    3. \(30\%\)
    4. \(45\%\)
    5. \(60\%\)

  9. In the diagram, three lines intersect at a point.

    Six angles surround the central point of intersection. The first, third, and fifth angles are each labelled 'x degrees', and the second, fourth, and sixth angles are each unmarked.

    What is the value of \(x\)?

    1. \(30\)
    2. \(45\)
    3. \(60\)
    4. \(90\)
    5. \(120\)

  10. Starting at 1:00 p.m., Jorge watched three movies. The first movie was 2 hours and 20 minutes long. He took a 20 minute break and then watched the second movie, which was 1 hour and 45 minutes long. He again took a 20 minute break and then watched the last movie, which was 2 hours and 10 minutes long. At what time did the final movie end?

    1. \(\mbox{6:45 p.m.}\)
    2. \(\mbox{7:15 p.m.}\)
    3. \(\mbox{7:35 p.m.}\)
    4. \(\mbox{7:55 p.m.}\)
    5. \(\mbox{8:15 p.m.}\)

Part B: Each correct answer is worth 6.

  1. Anna thinks of an integer.

    The integer that Anna is thinking of could be

    1. \(12\)
    2. \(14\)
    3. \(16\)
    4. \(21\)
    5. \(26\)

  2. Natalie and Harpreet are the same height. Jiayin’s height is 161 cm. The average (mean) of the heights of Natalie, Harpreet and Jiayin is 171 cm. What is Natalie’s height?

    1. \(161\mbox{ cm}\)
    2. \(166\mbox{ cm}\)
    3. \(176\mbox{ cm}\)
    4. \(183\mbox{ cm}\)
    5. \(191\mbox{ cm}\)

  3. The ratio of apples to bananas in a box is \(3\!:\!2\). The total number of apples and bananas in the box cannot be equal to

    1. \(40\)
    2. \(175\)
    3. \(55\)
    4. \(160\)
    5. \(72\)

  4. A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. The first four figures are as shown:

    The first figure is a single triangle. The second figure builds on the first figure by adding a second triangle that shares a side with the first triangle. The third figure builds on the second figure by adding a third triangle that shares a side with the second triangle. The fourth figure builds on the third figure by adding a fourth triangle that shares a side with the third triangle, but does not share a vertex with the first triangle.

    How many tiles are used to form the figure in the sequence with perimeter 91 cm?

    1. \(6\)
    2. \(11\)
    3. \(13\)
    4. \(15\)
    5. \(23\)

  5. In the diagram, the large square has area 49, the medium square has area 25, and the small square has area 9. The region inside the small square is shaded. The region between the large and medium squares is shaded.

    A small shaded square is centred within a medium unshaded square that is centred within a large shaded square.

    What is the total area of the shaded regions?

    1. \(33\)
    2. \(58\)
    3. \(45\)
    4. \(25\)
    5. \(13\)

  6. Which of the following expressions is not equivalent to \(3x+6\)?

    1. \(3(x+2)\)
    2. \(\dfrac{-9x-18}{-3}\)
    3. \(\dfrac{1}{3}(3x)+\dfrac{2}{3}(9)\)
    4. \(\dfrac{1}{3}(9x+18)\)
    5. \(3x-2(-3)\)

  7. Ben participates in a prize draw. He receives one prize that is equally likely to be worth $5, $10 or $20. Jamie participates in a different prize draw. She receives one prize that is equally likely to be worth $30 or $40. What is the probability that the total value of their prizes is exactly $50?

    1. \(\frac{1}{6}\)
    2. \(\frac{1}{3}\)
    3. \(\frac{1}{2}\)
    4. \(\frac{2}{5}\)
    5. \(\frac{2}{3}\)

  8. A positive integer \(n\) is a multiple of 7. The square root of \(n\) is between 17 and 18. How many possible values of \(n\) are there?

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

  9. Each of the following 15 cards has a letter on one side and a positive integer on the other side.

    There are three rows of five cards labelled as follows: First row is e, 17, 57, 60, D. Second row is 43, E, 3, 7, 13. Third row is 31, 88, G, H, 21.

    What is the minimum number of cards that need to be turned over to check if the following statement is true?

    “If a card has a lower case letter on one side, then it has an odd integer on the other side.”

    1. \(11\)
    2. \(9\)
    3. \(7\)
    4. \(5\)
    5. \(3\)

  10. A large \(5\times 5\times 5\) cube is formed using 125 small \(1\times 1\times 1\) cubes. There are three central columns, each passing through the small cube at the very centre of the large cube: one from top to bottom, one from front to back, and one from left to right. All of the small cubes that make up these three columns are removed. What is the surface area of the resulting solid?

    1. \(204\)
    2. \(206\)
    3. \(200\)
    4. \(196\)
    5. \(192\)

Part C: Each correct answer is worth 8.

  1. In the \(4\times 5\) grid shown, six of the \(1\times 1\) squares are not intersected by either diagonal.

    image

    When the two diagonals of an \(8\times 10\) grid are drawn, how many of the \(1\times 1\) squares are not intersected by either diagonal?

    1. \(44\)
    2. \(24\)
    3. \(52\)
    4. \(48\)
    5. \(56\)

  2. In the diagram, \(PQ\) is a diameter of a larger circle, point \(R\) is on \(PQ\), and smaller semi-circles with diameters \(PR\) and \(QR\) are drawn.

    The larger circle is drawn with a vertical diameter PR (P is at the top of the circle, and Q is at the bottom). R lies on PQ but is closer to Q. The right side of the larger circle is unshaded except for the shaded semi-circle with diameter QR. The left side of the larger circle is shaded except for the unshaded semi-circle with diameter PR.

    If \(PR=6\) and \(QR=4\), what is the ratio of the area of the shaded region to the area of the unshaded region?

    1. \(4:9\)
    2. \(2:3\)
    3. \(3:5\)
    4. \(2:5\)
    5. \(1:2\)

  3. Ali, Bea, Che, and Deb compete in a checkers tournament. Each player plays each other player exactly once. At the end of each game, either the two players tie or one player wins and the other player loses. A player earns 5 points for a win, 0 points for a loss, and 2 points for a tie. Exactly how many of the following final point distributions are possible?

    Player Points
    Ali 15
    Bea 7
    Che 4
    Deb 2
    Player Points
    Ali 10
    Bea 10
    Che 4
    Deb 4
    Player Points
    Ali 15
    Bea 5
    Che 5
    Deb 2
    Player Points
    Ali 12
    Bea 10
    Che 5
    Deb 0
    1. \(0\)
    2. \(1\)
    3. \(2\)
    4. \(3\)
    5. \(4\)

  4. Lucas chooses one, two or three different numbers from the list \(2,5,7,12,19,31,50,81\) and writes down the sum of these numbers. (If Lucas chooses only one number, this number is the sum.) How many different sums less than or equal to 100 are possible? 

    1. \(43\)
    2. \(39\)
    3. \(42\)
    4. \(40\)
    5. \(41\)

  5. We call the pair \((m,n)\) of positive integers a happy pair if the greatest common divisor of \(m\) and \(n\) is a perfect square. For example, \((20,24)\) is a happy pair because the greatest common divisor of 20 and 24 is 4. Suppose that \(k\) is a positive integer such that \((205\,800,35k)\) is a happy pair. The number of possible values of \(k\) with \(k \leq 2940\) is

    1. \(36\)
    2. \(28\)
    3. \(24\)
    4. \(30\)
    5. \(27\)

Further Information

For students...

Thank you for writing the Pascal Contest!

Encourage your teacher to register you for the Fryer Contest which will be written in April.

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to