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2016 Pascal Contest
(Grade 9)

Wednesday, February 24, 2016
(in North America and South America)

Thursday, February 25, 2016
(outside of North American and South America)

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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. Consider the following addition problem.

    300 plus 2020 plus 10001 equals.

    The result of the addition is

    1. \(15021\)
    2. \(12231\)
    3. \(12051\)
    4. \(13231\)
    5. \(12321\)

  2. Which of the following has the largest value?

    1. \(4^2\)
    2. \(4\times 2\)
    3. \(4-2\)
    4. \(\dfrac{4}{2}\)
    5. \(4+2\)

  3. In the diagram, the \(5 \times 6\) grid is made out of thirty \(1 \times 1\) squares. What is the total length of the six solid line segments shown?

    The six solid line segments have the following lengths: 5, 1, 4, 2, 3, and 3.

    1. \(6\)
    2. \(12\)
    3. \(16\)
    4. \(18\)
    5. \(20\)

  4. In the diagram, each of the five squares is \(1 \times 1\).

    Two squares are shaded and three squares are unshaded.

    What percentage of the total area of the five squares is shaded? 

    1. \(25\%\)
    2. \(30\%\)
    3. \(35\%\)
    4. \(40\%\)
    5. \(45\%\)

  5. Numbers \(m\) and \(n\) are on the number line, as shown.

    A number line with 16 equally spaced tick marks. 0 is the leftmost tick mark, 30 is six tick marks right of 0 and 60 is six tick marks right of 30. m is three ticks left of 0 and n is two ticks left of 60.

    The value of \(n-m\) is

    1. \(66\)
    2. \(35\)
    3. \(55\)
    4. \(60\)
    5. \(54\)

  6. If the symbol  \(\begin{array}{c|c} p & q \\ \hline r & s \\ \end{array}\)  is defined by \(p \times s - q \times r\), then the value of  \(\begin{array}{c|c} 4 & 5 \\ \hline 2 & 3 \\ \end{array}\)   is
    1. \(-3\)
    2. \(-2\)
    3. \(2\)
    4. \(3\)
    5. \(14\)

  7. Which of the following is equal to 2 m plus 3 cm plus 5 mm?

    1. \(2.035\mbox{ m}\)
    2. \(2.35\mbox{ m}\)
    3. \(2.0305\mbox{ m}\)
    4. \(2.53\mbox{ m}\)
    5. \(2.053\mbox{ m}\)

  8. If \(x=3\), \(y=2x\), and \(z=3y\), then the average of \(x\), \(y\) and \(z\) is

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

  9. A soccer team played three games. Each game ended in a win, loss, or tie. (If a game finishes with both teams having scored the same number of goals, the game ends in a tie.) In total, the team scored more goals than were scored against them. Which of the following combinations of outcomes is not possible for this team?

    1. \(\mbox{2 wins, 0 losses, 1 tie}\)
    2. \(\mbox{1 win, 2 losses, 0 ties}\)
    3. \(\mbox{0 wins, 1 loss, 2 ties}\)
    4. \(\mbox{1 win, 1 loss, 1 tie}\)
    5. \(\mbox{1 win, 0 losses, 2 ties}\)

  10. Exactly three faces of a \(2\times 2 \times 2\) cube are partially shaded, as shown. (Each of the three faces not shown in the diagram is not shaded.)

    On each of the three visible square faces a diagonal line is drawn from one corner to the opposite corner creating two triangles. One triangle is shaded and the other is left unshaded.

    What fraction of the total surface area of the cube is shaded?

    1. \(\frac{1}{3}\)
    2. \(\frac{1}{4}\)
    3. \(\frac{1}{6}\)
    4. \(\frac{3}{8}\)
    5. \(\frac{2}{3}\)

Part B: Each correct answer is worth 6.

  1. An oblong number is the number of dots in a rectangular grid with one more row than column. The first four oblong numbers are 2, 6, 12, and 20, and are represented below:

    Four grids of dots labelled first, second, third, and fourth. The first grid has 1 column with 2 dots. The second grid has 2 columns with 3 dots each. The third grid has 3 columns with 4 dots each. The fourth grid has 4 columns with 5 dots each.

    What is the 7th oblong number?

    1. \(42\)
    2. \(49\)
    3. \(56\)
    4. \(64\)
    5. \(72\)

  2. In the diagram, the area of square \(QRST\) is 36. Also, the length of \(PQ\) is one-half of the length of \(QR\).

    Rectangle PRSU with Q on PR and T on US. A line segment connects Q and T dividing rectangle PRSU into smaller rectangle PQTU and square QRST.

    What is the perimeter of rectangle \(PRSU\)?

    1. \(24\)
    2. \(30\)
    3. \(90\)
    4. \(45\)
    5. \(48\)

  3. Multiplying \(x\) by 10 gives the same result as adding 20 to \(x\). The value of \(x\) is

    1. \(\frac{9}{20}\)
    2. \(\frac{20}{9}\)
    3. \(\frac{11}{20}\)
    4. \(\frac{20}{11}\)
    5. \(2\)

  4. In the diagram, \(PQ\) is perpendicular to \(QR\), \(QR\) is perpendicular to \(RS\), and \(RS\) is perpendicular to \(ST\).

    Q is to the right of P, R is below Q, S is to the right of R, and T is above S. A dashed line passing through line segment QR joins P to T.

    If \(PQ=4\), \(QR=8\), \(RS=8\), and \(ST=3\), then the distance from \(P\) to \(T\) is

    1. \(16\)
    2. \(12\)
    3. \(17\)
    4. \(15\)
    5. \(13\)

  5. When two positive integers \(p\) and \(q\) are multiplied together, their product is 75. The sum of all of the possible values of \(p\) is

    1. \(96\)
    2. \(48\)
    3. \(109\)
    4. \(115\)
    5. \(124\)

  6. An integer from 10 to 99 inclusive is randomly chosen so that each such integer is equally likely to be chosen. The probability that at least one digit of the chosen integer is a 6 is

    1. \(\frac{1}{5}\)
    2. \(\frac{1}{10}\)
    3. \(\frac{1}{9}\)
    4. \(\frac{19}{90}\)
    5. \(\frac{19}{89}\)

  7. What is the tens digit of the smallest six-digit positive integer that is divisible by each of 10, 11, 12, 13, 14, and 15?

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

  8. Each integer from 1 to 12 is to be placed around the outside of a circle so that the positive difference between any two integers next to each other is at most \(2\). The integers 3, 4, \(x\), and \(y\) are placed as shown.

    Twelve positions are marked around a circle with eight positions empty. Moving around the circle, the numbers and variables are placed as follows: the number 3, empty position, empty position, the number 4, empty position, the variable x, empty position, then the variable y.

    What is the value of \(x + y\)?

    1. \(17\)
    2. \(18\)
    3. \(19\)
    4. \(20\)
    5. \(21\)

  9. Chris received a mark of \(50\%\) on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered \(25\%\) of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the test?

    1. \(23\)
    2. \(38\)
    3. \(32\)
    4. \(24\)
    5. \(40\)

  10. In the diagram, points \(Q\) and \(R\) lie on \(PS\) and \(\angle QWR = 38^\circ\).

    The point W lies above horizontal line segment PS. There are two additional points above PS: T to the left of W and V to the right of W. Also, a point U lies below PS.

    If \(\angle TQP = \angle TQW = x^\circ\), \(\angle VRS = \angle VRW = y^\circ\), and \(U\) is the point of intersection of \(TQ\) extended and \(VR\) extended, then the measure of \(\angle QUR\) is

    1. \(71^\circ\)
    2. \(45^\circ\)
    3. \(76^\circ\)
    4. \(81^\circ\)
    5. \(60^\circ\)



Part C: Each correct answer is worth 8.

  1. Grid lines are drawn on three faces of a rectangular prism, as shown.

    Two intersecting grid lines are drawn on the top face so the face looks like a two-by-two grid of unit squares. One grid line is drawn on the front face so that the face looks like two unit squares side by side. One grid line is drawn on the visible side face so that the face looks like two unit squares side by side. Point P is located at a corner of the top face, and point Q is located at a corner of the bottom face, diagonally opposite point P.

    A squirrel walks from \(P\) to \(Q\) along the edges and grid lines in such a way that she is always getting closer to \(Q\) and farther away from \(P\). How many different paths from \(P\) to \(Q\) can the squirrel take?

    1. \(14\)
    2. \(10\)
    3. \(20\)
    4. \(12\)
    5. \(16\)

  2. There are \(n\) students in the math club at Scoins Secondary School. When Mrs. Fryer tries to put the \(n\) students in groups of 4, there is one group with fewer than 4 students, but all of the other groups are complete. When she tries to put the \(n\) students in groups of 3, there are 3 more complete groups than there were with groups of 4, and there is again exactly one group that is not complete. When she tries to put the \(n\) students in groups of 2, there are 5 more complete groups than there were with groups of 3, and there is again exactly one group that is not complete. The sum of the digits of the integer equal to \(n^2-n\) is

    1. \(11\)
    2. \(12\)
    3. \(20\)
    4. \(13\)
    5. \(10\)

  3. In the diagram, \(\triangle PQR\) is isosceles with \(PQ = PR = 39\) and \(\triangle SQR\) is equilateral with side length 30.

    The point S is inside triangle PQR. Line segments join S to the three vertices of the triangle, dividing the triangle into three triangular pieces.

    The area of \(\triangle PQS\) is closest to

    1. \(68\)
    2. \(75\)
    3. \(50\)
    4. \(180\)
    5. \(135\)

  4. Ten very small rubber balls begin equally spaced inside a 55 m long tube. They instantly begin to roll inside the tube at a constant velocity of 1 m/s. When a ball reaches an end of the tube, it falls out of the tube. When two balls bump into each other, they both instantly reverse directions but continue to roll at 1 m/s. Five configurations giving the initial direction of movement of each ball are shown. All gaps indicated in the diagram are the same length and are equal in length to the distance from the ends of the tube to the nearest ball. For which configuration will it take the least time for more than half of the balls to fall out of the tube?

    1. The initial directions of movement of the 10 balls in order are: Right, Right, Right, Left, Right, Left, Left, Right, Left, Right.
    2. The initial directions are: Left, Right, Left, Right, Right, Right, Right, Left, Left, Left.
    3. The initial directions are: Right, Left, Left, Right, Right, Left, Left, Right, Left, Right.
    4. The initial directions are: Left, Right, Right, Left, Right, Left, Left, Left, Left, Right.
    5. The initial directions are: Right, Left, Right, Left, Right, Left, Left, Right, Left, Left.

  5. A 0 or 1 is to be placed in each of the nine \(1 \times 1\) squares in the \(3 \times 3\) grid shown so that each row contains at least one 0 and at least one 1, and each column contains at least one 0 and at least one 1.

    The number of ways in which this can be done is

    1. \(126\)
    2. \(120\)
    3. \(138\)
    4. \(102\)
    5. \(96\)

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.

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For teachers...

Visit our website cemc.uwaterloo.ca to