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2017 Cayley Contest
(Grade 10)

Tuesday, February 28, 2017
(in North America and South America)

Wednesday, March 1, 2017
(outside of North American and South America)

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©2017 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 \(6\times 111 - 2\times 111\) equals

    1. \(222\)
    2. \(333\)
    3. \(444\)
    4. \(555\)
    5. \(666\)

  2. The value of \(\dfrac{5^{2}-9}{5-3}\) is

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

  3. A snowman is built by stacking three spheres with their centres aligned vertically. The spheres have radii of 10 cm, 20 cm and 30 cm.

    4. How tall is the snowman? 

    1. \(\mbox{90 cm}\)
    2. \(\mbox{100 cm}\)
    3. \(\mbox{110 cm}\)
    4. \(\mbox{120 cm}\)
    5. \(\mbox{130 cm}\)

  4. Which of the following fractions has the greatest value?

    1. \(\frac{44444}{55555}\)
    2. \(\frac{5555}{6666}\)
    3. \(\frac{666}{777}\)
    4. \(\frac{77}{88}\)
    5. \(\frac{8}{9}\)

  5. The graph shows the volume of water in a 300 L tank as it is being drained at a constant rate.

    Line graph titled Volume of Water in Tank. X axis labelled Volume (litres). Y axis labelled Time (hours). The y intercept is 300 and the x intercept is 25.

    At what rate is the water leaving the tank, in litres per hour?

    1. \(12\)
    2. \(20\)
    3. \(2.5\)
    4. \(5\)
    5. \(15\)

  6. Penelope folds a piece of paper in half, creating two layers of paper. She folds the paper in half again, creating a total of four layers of paper. If she continues to fold the paper in half, which of the following is a possible number of layers that could be obtained?

    1. \(10\)
    2. \(12\)
    3. \(14\)
    4. \(16\)
    5. \(18\)

  7. The operation \(\Diamond\) is defined by \(a\,\Diamond\, b = a^{2}b-ab^{2}\). The value of \(2\,\Diamond\,7\) is

    1. \(-140\)
    2. \(-70\)
    3. \(0\)
    4. \(70\)
    5. \(140\)

  8. Each of three cards is labelled with three numbers. Which of the following groups of three cards has the properties that the first and second cards have exactly one number in common, the first and third cards have exactly one number in common, and the second and third cards have exactly one number in common?

    1. \(\fbox{\tt\,1\,3\,5}~\fbox{\tt\,3\,6\,7}~\fbox{\tt\,2\,4\,6}\)
    2. \(\fbox{\tt\,1\,4\,7}~\fbox{\tt\,2\,3\,4}~\fbox{\tt\,2\,4\,5}\)
    3. \(\fbox{\tt\,2\,3\,4}~\fbox{\tt\,2\,5\,7}~\fbox{\tt\,1\,2\,4}\)
    4. \(\fbox{\tt\,1\,4\,7}~\fbox{\tt\,2\,3\,4}~\fbox{\tt\,2\,5\,7}\)
    5. \(\fbox{\tt\,1\,3\,5}~\fbox{\tt\,1\,4\,7}~\fbox{\tt\,2\,3\,5}\)

  9. A restaurant bill, including 13% tax but not including a tip, is $226. The server is paid a tip of 15% based on the bill before tax. How much is the tip that the server is paid?

    1. \(\$32.87\)
    2. \(\$29.49\)
    3. \(\$30.00\)
    4. \(\$28.00\)
    5. \(\$44.07\)

  10. In the diagram, \(TU\) is parallel to \(PS\) and points \(Q\) and \(R\) lie on \(PS\).
    Also, \(\angle PQT = x^\circ\), \(\angle RQT = (x-50)^\circ\), and \(\angle TUR = (x+25)^\circ\).

    TURQ makes a trapezoid that sits on line PS.

    What is the measure of \(\angle URS\)?

    1. \(115^\circ\)
    2. \(140^\circ\)
    3. \(135^\circ\)
    4. \(130^\circ\)
    5. \(120^\circ\)

Part B: Each correct answer is worth 6.

  1. If \(a=\dfrac{2}{3}b\) and \(b \neq 0\), then \(\dfrac{9a+8b}{6a}\) is equal to

  2. The figure shown is made up of 10 identical squares.

    Two vertical stacks of 4 squares are joined by a horizontal row of 4 squares, creating a U shape.

    If the area of the figure is 160 \(\text{cm}^{2}\), what is the perimeter of the figure?

    1. \(72 \text{ cm}\)
    2. \(80\text{ cm}\)
    3. \(88\text{ cm}\)
    4. \(64\text{ cm}\)
    5. \(100\text{ cm}\)

  3. The mean (average) of the three integers \(p\), \(q\) and \(r\) is 9.
    The mean of the two integers \(s\) and \(t\) is 14.
    The mean of the five integers \(p\), \(q\), \(r\), \(s\), and \(t\) is

    1. \(11\)
    2. \(11.5\)
    3. \(12\)
    4. \(10\)
    5. \(13\)

  4. In the addition shown, each of \(X\), \(Y\) and \(Z\) represents a digit.

    Vertical addition problem set up with three numbers whose digits from left to right read XYZ, XYZ, and YZ. They sum to make 1675.

    What is the value of \(X+Y+Z\)?

    1. \(10\)
    2. \(15\)
    3. \(22\)
    4. \(20\)
    5. \(8\)

  5. Igor is shorter than Jie. Faye is taller than Goa. Jie is taller than Faye. Han is shorter than Goa. Who is the tallest?

    1. \(\mbox{Faye}\)
    2. \(\mbox{Goa}\)
    3. \(\mbox{Han}\)
    4. \(\mbox{Igor}\)
    5. \(\mbox{Jie}\)

  6. A bag contains red, blue and purple marbles, and does not contain any other marbles. The ratio of the number of red marbles to the number of blue marbles is \(4:7\). The ratio of the number of blue marbles to the number of purple marbles is \(2:3\). There are 32 red marbles in the bag. In total, how many marbles are there in the bag?

    1. \(162\)
    2. \(129\)
    3. \(176\)
    4. \(164\)
    5. \(172\)

  7. If \(x+2y = 30\), the value of \(\dfrac{x}{5}+\dfrac{2y}{3}+ \dfrac{2y}{5}+\dfrac{x}{3}\) is

    1. \(8\)
    2. \(16\)
    3. \(18\)
    4. \(20\)
    5. \(30\)

  8. The positive integers \(r\), \(s\) and \(t\) have the property that \(r\times s\times t = 1230\). What is the smallest possible value of \(r+s+t\)?

    1. \(51\)
    2. \(52\)
    3. \(54\)
    4. \(58\)
    5. \(53\)

  9. The number of integers \(n\) for which \(\dfrac{1}{7}\leq\dfrac{6}{n}\leq\dfrac{1}{4}\) is

    1. \(17\)
    2. \(18\)
    3. \(19\)
    4. \(20\)
    5. \(24\)

  10. Two lines with slopes \(\frac{1}{4}\) and \(\frac{5}{4}\) intersect at \((1,1)\). What is the area of the triangle formed by these two lines and the vertical line \(x=5\)?

    1. \(5\)
    2. \(10\)
    3. \(8\)
    4. \(12\)
    5. \(15\)

  11. Car X and Car Y are travelling in the same direction in two different lanes on a long straight highway. Car X is travelling at a constant speed of 90 km/h and has a length of 5 m. Car Y is travelling at a constant speed of 91 km/h and has a length of 6 m. Car Y starts behind Car X and eventually passes Car X. The length of time between the instant when the front of Car Y is lined up with the back of Car X and the instant when the back of Car Y is lined up with the front of Car X is \(t\) seconds. The valueof \(t\) is

    1. \(39.6\)
    2. \(18.0\)
    3. \(21.6\)
    4. \(46.8\)
    5. \(32.4\)

Part C: Each correct answer is worth 8.

  1. The integers 1 to 6 are to be inserted into the grid shown.

    6 squares arranged into two vertical stacks of three. The right stack is higher than the left stack by one square so that the first square of the left stack touches the second square of the right stack. The bottom square of the first stack is labelled x. The top square of the right stack is labelled 1.

    No two integers that differ by 1 may be in squares that share an edge. If the 1 is inserted as shown, how many different integers can be placed in the box labelled \(x\)?

    1. \(252\)
    2. \(432\)
    3. \(441\)
    4. \(490\)
    5. \(540\)

  2. In the diagram, square \(PQRS\) has side length 42 and is divided into four non-overlapping rectangles.

    Square PQRS is divided by a line running parallel to RS into a larger and smaller section. The larger section is divided by a line parallel to QR into a larger and smaller section. The new larger section is split into two equal sections by a line parallel to RS. One of these sections is shaded.

    If each of these four rectangles has the same perimeter, what is the area of the shaded rectangle?

    1. \(252\)
    2. \(432\)
    3. \(441\)
    4. \(490\)
    5. \(540\)

  3. The triangle with side lengths 6, 8 and 10 is right-angled, while the triangle with side lengths 6, 8 and 9 is an acute triangle and the triangle with side lengths 6, 8 and 11 is an obtuse triangle. An obtuse triangle with positive area has side lengths 10, 17 and \(x\). If \(x\) is an integer, what is the sum of all possible values of \(x\)?

    1. \(161\)
    2. \(198\)
    3. \(63\)
    4. \(323\)
    5. \(224\)

  4. Three coins are placed in the first three of six squares, as shown.

         The first set of six squares, labelled start has one coin in each of the first three squares, labelled x y and z respectively. The 6 squares labelled finish have the coins in the same order in the last three squares.

    A move consists of moving one coin one space to the right, assuming that this space is empty. (No coin can jump over another coin, so the order of the coins will never change.) How many different sequences of moves can be used to move the three coins from the first three squares to the last three squares?

    1. \(44\)
    2. \(40\)
    3. \(42\)
    4. \(48\)
    5. \(50\)

  5. A positive integer \(n\) with \(n \geq 3\) is called a Nella number if there exists a positive integer \(x\) with \(x<n\) and there exists a positive integer \(m\) such that

    For example, \(n=7\) is a Nella number. How many Nella numbers \(n\) are there with \(50 \leq n \leq 2017\)?

    1. \(393\)
    2. \(394\)
    3. \(395\)
    4. \(396\)
    5. \(397\)

Further Information

For students...

Thank you for writing the Cayley Contest!

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

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to