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2015 Pascal Contest
(Grade 9)

Tuesday, February 24, 2015
(in North America and South America)

Wednesday, February 25, 2015
(outside of North American and South America)

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©2014 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 value of \(\dfrac{20+15}{30-25}\) is

    1. \(1\)
    2. \(5\)
    3. \(2\)
    4. \(7\)
    5. \(0\)

  2. Which of the following figures is obtained when the shaded figure shown is reflected about the line segment \(PQ\)?
        A horizontal line PQ with a 5-sided shape above it. The 5 sided shape is a rectangle with the bottom side concaving to make it 5 sides.

    1. A horizontal line PQ with a 5-sided shape below it. The 5 sided shape is a rectangle with the top side concaving to make it 5 sides.
    2. A horizontal line PQ with a 5-sided shape below it. The 5 sided shape is a rectangle with the bottom side concaving to make it 5 sides.
    3. A horizontal line PQ with a rectangle below it. The rectangle convexes past PQ.
    4. A horizontal line PQ with a 5-sided shape below it. The 5 sided shape is a rectangle with the left side concaving to make it 5 sides.
    5. A horizontal line PQ with a 5-sided shape above it. The 5 sided shape is a rectangle with the left side concaving to make it 5 sides.

  3. If \(8 + 6 = n + 8\), then \(n\) equals

    1. \(14\)
    2. \(22\)
    3. \(6\)
    4. \(-2\)
    5. \(9\)

  4. Which of the following numbers is greater than 0.7?

    1. \(0.07\)
    2. \(-0.41\)
    3. \(0.8\)
    4. \(0.35\)
    5. \(-0.9\)

  5. The expression \(4 + \frac{3}{10} + \frac{9}{1000}\) is equal to

    1. \(4.12\)
    2. \(4.309\)
    3. \(4.039\)
    4. \(4.012\)
    5. \(4.39\)

  6. The average age of Andras, Frances and Gerta is 22 years.

    Name Age (Years)
    Andras 23
    Frances 24
    Gerta ?
    What is Gerta’s age?

    1. \(19\)
    2. \(20\)
    3. \(21\)
    4. \(22\)
    5. \(23\)

  7. If \(n = 7\), which of the following expressions is equal to an even integer?

    1. \(9n\)
    2. \(n+8\)
    3. \(n^2\)
    4. \(n(n-2)\)
    5. \(8n\)

  8. Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail?

    1. \(28\mbox{ km}\)
    2. \(12.8\mbox{ km}\)
    3. \(11.2\mbox{ km}\)
    4. \(13\frac{1}{3}\mbox{ km}\)
    5. \(20\mbox{ km}\)

  9. In the diagram, line segments \(PQ\) and \(RS\) intersect at \(T\).

    Two triangles SPT and QRT are conjoined at point T. QRT is a right angle triangle with R being a right angle and Q being 50 degrees on the inside of the triangle. For SPT, P measures 110 degrees and S measures x degrees.

    The value of \(x\) is

    1. \(30\)
    2. \(20\)
    3. \(40\)
    4. \(50\)
    5. \(35\)

  10. The value of \(\sqrt{16 \times \sqrt{16}}\) is

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

Part B: Each correct answer is worth 6.

  1. Jim wrote the sequence of symbols \(\heartsuit\,\spadesuit\,\spadesuit\,\heartsuit\,\diamondsuit\,\heartsuit\,\heartsuit\,\heartsuit\,\diamondsuit\) a total of 50 times. How many more \(\heartsuit\) symbols than \(\spadesuit\) symbols did he write?

    1. \(50\)
    2. \(150\)
    3. \(200\)
    4. \(250\)
    5. \(275\)

  2. What is the smallest positive integer that is a multiple of each of 3, 5, 7, and 9?

    1. \(35\)
    2. \(105\)
    3. \(210\)
    4. \(315\)
    5. \(630\)

  3. Sixteen squares are arranged to form a region, as shown.

    The region is a 5 by 5 grid with the four corner squares removed. The outer edge of the region is outlined with a solid line. The middle three squares in the third row along with the middle three squares in the third column form a plus sign at the center of the grid. This plus sign is outlined with dotted line.

    Each square has an area of 400 \(\mbox{m}^2\). Anna walks along the solid line path formed by the outer edge of the region exactly once. Aaron walks along the dotted line path formed by the inner edges of the region exactly once. In total, how far did Anna and Aaron walk?

    1. \(160\mbox{ m}\)
    2. \(240\mbox{ m}\)
    3. \(320\mbox{ m}\)
    4. \(400\mbox{ m}\)
    5. \(640\mbox{ m}\)

  4. The operation \(\otimes\) is defined by \(a \otimes b = \dfrac{a}{b}+\dfrac{b}{a}\). What is the value of \(4 \otimes 8\)?

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

  5. At the end of the year 2000, Steve had $100 and Wayne had $10 000. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for the first time?

    1. \(2002\)
    2. \(2003\)
    3. \(2004\)
    4. \(2005\)
    5. \(2006\)

  6. Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton.

    A number line between the endpoints Mathville and Staton. The distance between the two endpoints is 200 kilometers.

    Bruce drove at 50 km/h. Anca drove at 60 km/h, but stopped along the way to rest. They both arrived at Staton at the same time. For how long did Anca stop to rest?

    1. \(40\mbox{ minutes}\)
    2. \(10\mbox{ minutes}\)
    3. \(67\mbox{ minutes}\)
    4. \(33\mbox{ minutes}\)
    5. \(27\mbox{ minutes}\)

  7. In the diagram, six identical circles just touch the edges of rectangle \(PQRS\) and each circle just touches the adjacent circles. The centres \(T, V, W, Y\) of four of these circles form a smaller rectangle \(TVWY\), as shown.

    The centres \(U\) and \(X\) lie on this rectangle. If the perimeter of \(TVWY\) is 60, what is the area of \(PQRS\)?

    1. \(600\)
    2. \(900\)
    3. \(400\)
    4. \(1200\)
    5. \(1000\)

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

    \[\begin{array}{|c|c|c|} \hline a & 13 & b \\ \hline 19 & c & 11 \\ \hline 12 & d & 16 \\ \hline \end{array}\]

    In the magic square shown, the sum \(a+b+c\) equals

    1. \(49\)
    2. \(54\)
    3. \(47\)
    4. \(50\)
    5. \(46\)

  9. Krystyna has some raisins. She gives one-third of her raisins to Mike. She then eats 4 raisins, after which she gives one-half of her remaining raisins to Anna. If Krystyna then has 16 raisins left, how many raisins did she have to begin?

    1. \(42\)
    2. \(54\)
    3. \(60\)
    4. \(84\)
    5. \(108\)

  10. André has an unlimited supply of $1 coins, $2 coins, and $5 bills. Using only these coins and bills and not necessarily using some of each kind, in how many different ways can he form exactly $10?

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

Part C: Each correct answer is worth 8.

  1. Each diagram shows a triangle, labelled with its area.

    Three triangles with their vertices as coordinates on a graph. The first triangle's vertices are (0,0), (1,4), and (4,1) with its area labelled m. The first triangle's vertices are (0,1), (3,0), and (4,4) with its area labelled n. The first triangle's vertices are (0,4), (2,0), and (4,3) with its area labelled p.

    What is the correct ordering of the areas of these triangles?

    1. \(m<n<p\)
    2. \(p<n<m\)
    3. \(n<m<p\)
    4. \(n<p<m\)
    5. \(p<m<n\)

  2. The chart shown gives the cost of installing carpet in four rectangular rooms of various sizes. The cost per square metre of installing carpet is always the same.

    Width (metres)
    \(10\) \(y\)
    Length (metres) \(15\) \(\$ 397.50\) \(\$ 675.75\)
    \(x\) \(\$ 742.00\) \(z\)

    What is the value of \(z\)?

    1. \(331.25\)
    2. \(463.75\)
    3. \(1815.25\)
    4. \(476.00\)
    5. \(1261.40\)

  3. How many triples \((a, b, c)\) of positive integers satisfy the conditions \(6ab = c^2\) and \(a < b < c \leq 35\)?

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

  4. Paula, Quinn, Rufus, and Sarah are suspects in a crime. The police found links between exactly four pairs of suspects: Paula and Quinn, Quinn and Rufus, Rufus and Paula, and Quinn and Sarah. These links can be shown in a diagram by drawing a point to represent each suspect and a line or curve joining two points whenever the two corresponding suspects are linked. An example of a drawing that represents this information is:

    Points P, Q, and R are joined by lines to form a triangle. A point S lies outside the triangle and a line joins S to point Q.

    Ali, Bob, Cai, Dee, Eve, and Fay are suspects in a second crime. The police found links between exactly eight pairs of suspects: Ali and Bob, Bob and Cai, Cai and Dee, Dee and Eve, Eve and Fay, Fay and Ali, Ali and Dee, and Bob and Eve. For how many of the following drawings can the six dots be labelled with the names of the six suspects so that each of the eight links given is represented by a line or curve in that drawing?

    Five drawings consisting of six dots with some pairs of dots joined by a line or a curve. A description of the drawings follows.

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

  5. The first four rows of a table with columns \(V\), \(W\), \(X\), \(Y\), and \(Z\) are shown.

    \(V\) \(W\) \(X\) \(Y\) \(Z\)
    1 3 4 6 8
    2 5 7 11 15
    9 19 28 46 64
    10 21 31 51 71

    For each row, whenever integer \(n\) appears in column \(V\), column \(W\) contains the integer \(2n + 1\), column \(X\) contains \(3n + 1\), column \(Y\) contains \(5n+1\), and column \(Z\) contains \(7n+1\). For every row after the first, the number in column \(V\) is the smallest positive integer that does not yet appear in any previous row. The integer 2731 appears in column \(W\). The complete list of columns in which 2731 appears is

    1. \(\mbox{$W$}\)
    2. \(\mbox{$W$, $X$, $Y$, and $Z$}\)
    3. \(\mbox{$W$, $X$ and $Z$}\)
    4. \(\mbox{$W$, $Y$ and $Z$}\)
    5. \(\mbox{$W$ and $Z$}\)

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