CEMC Banner

2015 Cayley Contest
(Grade 10)

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

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

University of Waterloo Logo

©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 \(2\times 2015 - 2015\) is

    1. \(2015\)
    2. \(4030\)
    3. \(6045\)
    4. \(0\)
    5. \(-2015\)

  2. The expression \(\sqrt{1}+\sqrt{9}\) is equal to

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

  3. The base of a rectangular box measures 2 cm by 5 cm. The volume of the box is \(30\mbox{ cm}^3\). What is the height of the box?

    1. \(1\mbox{ cm}\)
    2. \(2\mbox{ cm}\)
    3. \(3\mbox{ cm}\)
    4. \(4\mbox{ cm}\)
    5. \(5\mbox{ cm}\)

  4. In the diagram, \(R\) lies on line segment \(PS\).

    A triangle with the vertices P, Q, and R. Line PR extends to point S to create line PS. Angle P measures 50 degrees on the inside, angle Q is a right angle, and angle QRS measures x degrees.


    The value of \(x\) is

    1. \(120\)
    2. \(130\)
    3. \(135\)
    4. \(140\)
    5. \(150\)

  5. The bar graph shows the number of provinces and territories that joined Canadian Confederation during each of four 40 year time periods.

    A bar graph with the x-axis title Time Period and labels 1850 to 1889, 1890 to 1929, 1930 to 1969, and 1970 to 2009. It has the y-axis title Number of Provinces and Territories and the scale shows 2, 4, 6, and 8 with ticks in between each of these increments. The 1850 to 1889 bar goes up to 8, the 1890 to 1929 bar goes up to the tick between 2 and 3, and the 1930 to 1969 and the 1970 to 2009 bars both go up to the tick between the x-axis and 2.

    If one of the 13 provinces or territories is chosen at random, what is the probability that it joined Canadian Confederation between 1890 and 1969?

    1. \(\frac{12}{13}\)
    2. \(\frac{4}{13}\)
    3. \(\frac{5}{13}\)
    4. \(\frac{3}{13}\)
    5. \(\frac{2}{13}\)

  6. If \(a^2 = 9\), then \(a^4\) equals

    1. \(27\)
    2. \(81\)
    3. \(243\)
    4. \(729\)
    5. \(2187\)

  7. The expression \(3 + \frac{1}{10} + \frac{4}{100}\) is not equal to

    1. \(3 \frac{14}{100}\)
    2. \(3.14\)
    3. \(3 \frac{5}{110}\)
    4. \(3 \frac{7}{50}\)
    5. \(\frac{157}{50}\)

  8. Violet has one-half of the money she needs to buy her mother a necklace. After her sister gives her $30, she has three-quarters of the amount she needs. Violet’s father agrees to give her the rest. The amount that Violet’s father will give her is

    1. \(\$7.50\)
    2. \(\$15\)
    3. \(\$22.50\)
    4. \(\$30\)
    5. \(\$120\)

  9. John goes for a jog every 3 days. He went for a jog on Monday, January 5. He went for his next jog on January 8. What was the date of the next Monday on which he went for a jog?

    1. \(\mbox{January 12}\)
    2. \(\mbox{January 19}\)
    3. \(\mbox{January 26}\)
    4. \(\mbox{February 2}\)
    5. \(\mbox{February 9}\)

  10. In the diagram, square \(PQRS\) is \(3\times 3\). Points \(T\) and \(U\) are on side \(QR\) with \(QT=TU=UR=1\). Points \(V\) and \(W\) are on side \(RS\) with \(RV=VW=WS=1\). Line segments \(TX\) and \(UY\) are perpendicular to \(QR\) and line segments \(VY\) and \(WX\) are perpendicular to \(RS\).

    The ratio of the shaded area to the unshaded area is

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

Part B: Each correct answer is worth 6.

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

  2. The points \((-1, q)\) and \((-3, r)\) are on a line parallel to \(y = \tfrac{3}{2}x + 1\). What is the value of \(r - q\)?

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

  3. At Barker High School, a total of 36 students are on either the baseball team, the hockey team, or both. If there are 25 students on the baseball team and 19 students on the hockey team, how many students play both sports?

    1. \(7\)
    2. \(8\)
    3. \(9\)
    4. \(10\)
    5. \(11\)

  4. In the diagram, \(\triangle PQR\) is isosceles with \(PQ=PR\) and \(\triangle PRS\) is isosceles with \(PS=SR=x\).

    Also, the perimeter of \(\triangle PQR\) is 22, the perimeter of \(\triangle PRS\) is 22, and the perimeter of \(PQRS\) is 24. What is the value of \(x\)?

    1. \(7.5\)
    2. \(6.5\)
    3. \(7\)
    4. \(6\)
    5. \(8\)

  5. If \(n\) is a positive integer, the symbol \(n!\) (read “\(n\) factorial") represents the product of the integers from \(1\) to \(n\). For example, \(4!=(1)(2)(3)(4)\) or \(4!=24\). The ones (units) digit of the sum \(1!+2!+3!+4!+5!+6!+7!+8!+9!+10!\) is

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

  6. 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. \(46\)

  7. For the first 30 minutes of a trip, Deanna drove at a constant speed. For the next 30 minutes, she drove at a constant speed 20 km/h faster than her original speed. If the total distance that she travelled was 100 km, how fast did she drive for the first 30 minutes?

    1. \(80\mbox{ km/h}\)
    2. \(90\mbox{ km/h}\)
    3. \(100\mbox{ km/h}\)
    4. \(110\mbox{ km/h}\)
    5. \(120\mbox{ km/h}\)

  8. In the diagram, rectangle \(PQRS\) has side \(PQ\) on the diameter of the semicircle with \(R\) and \(S\) on the semicircle.

    If the diameter of the semicircle is 20 and the length of \(PQ\) is 16, then the length of \(PS\) is

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

  9. A bank teller has some stacks of bills. The total value of the bills in each stack is $1000. Every stack contains at least one $20 bill, at least one $50 bill, and no other types of bills. If no two stacks have the same number of $20 bills, what is the maximum possible number of stacks that the teller could have?

    1. \(9\)
    2. \(10\)
    3. \(11\)
    4. \(4\)
    5. \(8\)

  10. For how many integers \(n\) is \(72\left(\frac{3}{2}\right)^n\) equal to an integer?

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

Part C: Each correct answer is worth 8.

  1. The average of a list of three consecutive odd integers is 7. When a fourth positive integer, \(m\), different from the first three, is included in the list, the average of the list is an integer. What is the sum of the three smallest possible values of \(m\)?

    1. \(6\)
    2. \(9\)
    3. \(21\)
    4. \(29\)
    5. \(33\)

  2. Six players compete in a chess tournament. Each player plays exactly two games against every other player. In each game, the winning player earns 1 point and the losing player earns 0 points; if the game results in a draw (tie), each player earns \(\frac{1}{2}\) point. What is the minimum possible number of points that a player needs to earn in order to guarantee that he has more points than every other player?

    1. \(8\)
    2. \(8\frac{1}{2}\)
    3. \(9\)
    4. \(9\frac{1}{2}\)
    5. \(10\)

  3. Nylah has her living room lights on a timer. Each evening, the timer switches the lights on randomly at exactly 7:00 p.m., 7:30 p.m., 8:00 p.m., 8:30 p.m., or 9:00 p.m. Later in the evening, the timer switches the lights off at any random time between 11 p.m. and 1 a.m. For example, the lights could be switched on at exactly 7:30 p.m. and off at any one of the infinite number of possible times between 11 p.m. and 1 a.m. On a given night, Nylah’s lights are on for \(t\) hours. What is the probability that \(4 < t < 5\)? 

    1. \(\frac{1}{2}\)
    2. \(\frac{1}{4}\)
    3. \(\frac{2}{5}\)
    4. \(\frac{3}{10}\)
    5. \(\frac{7}{20}\)

  4. In the diagram, a rectangular ceiling \(PQRS\) measures 6 m by 4 m and is to be completely covered using 12 rectangular tiles, each measuring 1 m by 2 m. Also, there is a beam, \(TU\), that is positioned so that \(PT=SU=2\mbox{ m}\) and that cannot be crossed by any tile.

    Rectangle PQRS with point T on top side PQ, closer to P, and point U on bottom side SR, closer to S. Smaller rectangle PTUS, a portion of PQRS, is covered by two tiles placed vertically side by side along with two tiles placed horizontally. Square TQRU, the other portion of PQRS, is covered by two tiles placed vertically and six tiles placed horizontally.

    The number of possible arrangements of tiles is

    1. \(180\)
    2. \(190\)
    3. \(185\)
    4. \(170\)
    5. \(175\)

  5. Rectangular prism \(PQRSWTUV\) has a square base \(PQRS\). Point \(X\) is on the face \(TUVW\) so that \(PX=12\), \(QX=10\) and \(RX=8\).

    A rectangular prism with bottom face PQRS with top face UVWT. Vertices, U, V, W, and T are above vertices P, Q, R, and S respectively.

    The maximum possible area of rectangle \(PQUT\) is closest to

    1. \(67.84\)
    2. \(67.82\)
    3. \(67.90\)
    4. \(67.86\)
    5. \(67.88\)

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