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2019 Pascal Contest
(Grade 9)

Tuesday, February 26, 2019
(in North America and South America)

Wednesday, February 27, 2019
(outside of North American and South America)

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©2018 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. The expression \(2\times 3+2\times 3\) equals

    1. \(10\)
    2. \(20\)
    3. \(36\)
    4. \(12\)
    5. \(16\)

  2. The perimeter of a square is 28. What is the side length of this square?

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

  3. In the diagram, some of the hexagons are shaded.

    There are a total of 9 regular hexagons in a diamond formation with 5 of the hexagons shaded and the other 4 unshaded.

    4. What fraction of all of the hexagons are shaded?

    1. \(\frac{1}{2}\)
    2. \(\frac{5}{9}\)
    3. \(\frac{4}{5}\)
    4. \(\frac{1}{3}\)
    5. \(\frac{5}{6}\)

  4. Yesterday, each student at Pascal C.I. was given a snack. Each student received either a muffin, yogurt, fruit, or a granola bar. No student received more than one of these snacks. The percentages of the students who received each snack are shown in the circle graph.

    A circle graph with the following percentages: 38% Muffin, 27% Fruit, 25% Granola Bar, and 10% Yogurt.

    5. What percentage of students did not receive a muffin?

    1. \(27\%\)
    2. \(38\%\)
    3. \(52\%\)
    4. \(62\%\)
    5. \(78\%\)

  5. What is the smallest integer that can be placed in the box so that \(\dfrac{1}{2}<\dfrac{\square}{9}\) ?

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

  6. If \(4x + 14 = 8x - 48\), what is the value of \(2x\)?

    1. \(17\)
    2. \(31\)
    3. \(35\)
    4. \(24\)
    5. \(36\)

  7. In the diagram, point \(P\) is on the number line at 3 and \(V\) is at 33.

    The number line between \(3\) and \(33\) is divided into six equal parts by the points \(Q,R,S,T,U\).

    What is the sum of the lengths of \(PS\) and \(TV\)?

    1. \(25\)
    2. \(23\)
    3. \(24\)
    4. \(21\)
    5. \(27\)

  8. The median of the numbers in the list \(19^{20},\dfrac{20}{19},20^{19},2019,20\times 19\) is

    1. \(19^{20}\)
    2. \(\dfrac{20}{19}\)
    3. \(20^{19}\)
    4. \(2019\)
    5. \(20\times 19\)

  9. In the diagram, each partially shaded circle has a radius of 1 cm and has a right angle marked at its centre.

    There are a total of 12 circles in a 3 by 4 formation with each circle having 3 quarters of it shaded.

    10. In \(\mbox{cm}^2\), what is the total shaded area?

    1. \(4\pi^2\)
    2. \(9\pi^2\)
    3. \(4\pi\)
    4. \(9\pi\)
    5. \(3\pi\)

  10. Three \(1 \times 1 \times 1\) cubes are joined face to face in a single row and placed on a table, as shown.

    11. The cubes have a total of 11 exposed \(1 \times 1\) faces. If sixty \(1 \times 1 \times 1\) cubes are joined face to face in a single row and placed on a table, how many \(1 \times 1\) faces are exposed?

    1. \(125\)
    2. \(220\)
    3. \(182\)
    4. \(239\)
    5. \(200\)

Part B: Each correct answer is worth 6.

  1. In a magic square, the numbers in each row, the numbers in each column, and the numbers on each diagonal have the same sum.

    A 3 by 3 grid with 9 cells. Going left to right and top to bottom, the first row first column cell is 2.3, the second row first column cell is 3.6, the second row second column cell is 3, the second row third column cell is 2.4, and finally the third row second column cell is x. The rest of the cells are blank.

    In the magic square shown, the value of \(x\) is

    1. \(3.8\)
    2. \(3.6\)
    3. \(3.1\)
    4. \(2.9\)
    5. \(2.2\)

  2. In the diagram, \(PR\) and \(QS\) meet at \(X\).

    An hourglass shaped figure with the vertices P, Q, R, and S. The point at the centre of the two triangles is labelled X.

    3. Also, \(\triangle PQX\) is right-angled at \(Q\) with \(\angle QPX = 62^\circ\) and \(\triangle RXS\) is isosceles with \(RX=SX\) and \(\angle XSR = y^\circ\). The value of \(y\) is

    1. \(54\)
    2. \(71\)
    3. \(76\)
    4. \(59\)
    5. \(60\)

  3. The list \(p,q,r,s\) consists of four consecutive integers listed in increasing order. If \(p+s=109\), the value of \(q+r\) is

    1. \(108\)
    2. \(109\)
    3. \(110\)
    4. \(117\)
    5. \(111\)

  4. Many of the students in M. Gamache’s class brought a skateboard or a bicycle to school yesterday. The ratio of the number of skateboards to the number of bicycles was \(7:4\). There were 12 more skateboards than bicycles. How many skateboards and bicycles were there in total?

    1. \(44\)
    2. \(33\)
    3. \(11\)
    4. \(22\)
    5. \(55\)

  5. Sophie has written three tests. Her marks were 73%, 82% and 85%. She still has two tests to write. All tests are equally weighted. Her goal is an average of 80% or higher. With which of the following pairs of marks on the remaining tests will Sophie not reach her goal?

    1. \(79\%\text{ and }82\%\)
    2. \(70\%\text{ and }91\%\)
    3. \(76\%\text{ and }86\%\)
    4. \(73\%\text{ and }83\%\)
    5. \(61\%\text{ and }99\%\)

  6. If \(x\) is a number less than \(-2\), which of the following expressions has the least value?

    1. \(x\)
    2. \(x+2\)
    3. \(\frac{1}{2}x\)
    4. \(x-2\)
    5. \(2x\)

  7. Hagrid has 100 animals. Among these animals,

    How many of Hagrid’s spotted animals have horns?

    1. \(8\)
    2. \(10\)
    3. \(2\)
    4. \(38\)
    5. \(26\)

  8. In the diagram, each of \(\triangle QPT\), \(\triangle QTS\) and \(\triangle QSR\) is an isosceles, right-angled triangle, with \(\angle QPT = \angle QTS = \angle QSR = 90^\circ\).

    Triangle QPT is on the left with vertex P sticking out on the left end, triangle QTS is just to its right with side TS on the bottom, and triangle QSR is on the right of QTS with vertex R sticking out on the top right corner and side QR on the top.

    9. The combined area of the three triangles is 56. If \(QP = PT = k\), what is the value of \(k\)?

    1. \(\sqrt{2}\)
    2. \(1\)
    3. \(4\)
    4. \(2\)
    5. \(2\sqrt{2}\)

  9. There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. How many different-looking arrangements are possible?

    1. \(15\)
    2. \(16\)
    3. \(10\)
    4. \(11\)
    5. \(12\)

  10. In the diagram, square \(PQRS\) has side length 40. Points \(J\), \(K\), \(L\), and \(M\) are on the sides of \(PQRS\), as shown, so that \(JQ = KR=LS=MP=10\).

    A square with the vertices P, Q, R, and S, starting from the top left corner and going clockwise. Point J is on the side of PQ, point K is on the side of QR, point L is on the side of RS, and the point M is on the side of SP. There is a diamond in the middle of the square with vertices W, X, Y, and Z. Vertex W is on the line JZ, vertex X is on the line KW, vertex Y is on the line LX, vertex Z is on the line MY.

    11. Line segments \(JZ\), \(KW\), \(LX\), and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)?

    1. \(280\)
    2. \(200\)
    3. \(320\)
    4. \(240\)
    5. \(160\)

Part C: Each correct answer is worth 8.

  1. What is the units (ones) digit of the integer equal to \(5^{2019}-3^{2019}\)?

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

  2. The integer 2019 can be formed by placing two consecutive two-digit positive integers, 19 and 20, in decreasing order. What is the sum of all four-digit positive integers greater than 2019 that can be formed in this way?

    1. \(476\,681\)
    2. \(476\,861\)
    3. \(478\,661\)
    4. \(468\,671\)
    5. \(468\,761\)

  3. A path of length 14 m consists of 7 unshaded stripes, each of length 1 m, alternating with 7 shaded stripes, each of length 1 m. A circular wheel of radius 2 m is divided into four quarters which are alternately shaded and unshaded. The wheel rolls at a constant speed along the path from the starting position shown.

    A circle on the left with the top left quarter and bottom left quarter shaded with a rectangle beneath it which is split into even smaller rectangles, alternately shaded. To the right end of this rectangle is 3 dots and there is also an arrow pointing to the right, above this rectangle.

    The wheel makes exactly 1 complete revolution. The percentage of time during which a shaded section of the wheel is touching a shaded part of the path is closest to

    1. \(20\%\)
    2. \(18\%\)
    3. \(16\%\)
    4. \(24\%\)
    5. \(22\%\)

  4. If \(p\), \(q\), \(r\), and \(s\) are digits, how many of the 14-digit positive integers of the form \(88\,663\,311\,pqr\,s48\) are divisible by 792?

    1. \(48\)
    2. \(56\)
    3. \(40\)
    4. \(60\)
    5. \(50\)

  5. In the diagram, \(PR\) and \(QS\) intersect at \(V\). Also, \(W\) is on \(PV\), \(U\) is on \(PS\) and \(T\) is on \(PQ\) with \(QU\) and \(ST\) passing through \(W\).

    A quadrilateral with the vertices P, Q, R, and S. Point T lies on line PQ and point U lies on line PS.

    6. For some real number \(x\),

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

    1. \(35\)
    2. \(34\)
    3. \(33\)
    4. \(32\)
    5. \(31\)

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