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2024 Cayley Contest
(Grade 10)

Wednesday, February 28, 2024
(in North America and South America)

Thursday, February 29, 2024
(outside of North American and South America)

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©2023 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. Part A and Part B of this contest are multiple choice. Each of the questions in these parts 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. The correct answer to each question in Part C is an integer from 0 to 99, inclusive. After deciding on your answer, fill in the appropriate two circles on the response form. A one-digit answer (such as "7") must be coded with a leading zero ("07").
  8. 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.
  9. Diagrams are not drawn to scale. They are intended as aids only.
  10. When your supervisor tells you to begin, you will have sixty minutes of working time.
  11. 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 \(2 \times 0 + 2 \times 4\) is equal to

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

  2. If \(x = 3\), the value of \(-(5x - 6x)\) is

    1. \(-33\)
    2. \(3\)
    3. \(-1\)
    4. \(33\)
    5. \(11\)

  3. In \(\triangle ABC\), points \(E\) and \(F\) are on \(AB\) and \(BC\), respectively, such that \(AE=BF\) and \(BE=CF\).

    If \(\angle BAC = 70\degree\), the measure of \(\angle ABC\) is

    1. \(40\degree\)
    2. \(50\degree\)
    3. \(60\degree\)
    4. \(70\degree\)
    5. \(30\degree\)

  4. At Wednesday’s basketball game, the Cayley Comets scored \(90\) points.
    At Friday’s game, they scored \(80\%\) as many points as they scored on Wednesday.
    How many points did they score on Friday?

    1. \(60\)
    2. \(72\)
    3. \(75\)
    4. \(78\)
    5. \(82\)

  5. In the diagram, the two identical bases of the prism are shaped like a star.

    The area of each star-shaped base is \(400 \text{ cm}^2\). The depth of the prism (that is, the distance between the star-shaped bases) is \(8\) cm. The volume of the prism is

    1. \(720\text{ cm}^3\)
    2. \(1520\text{ cm}^3\)
    3. \(3200\text{ cm}^3\)
    4. \(3600\text{ cm}^3\)
    5. \(28\,800\text{ cm}^3\)

  6. Last year, Lloyd ate cookies in the percentages shown in the pie chart.

    There are four types of cookies. Chocolate chip makes up 33%, oatmeal makes up 22%, and gingerbread and sugar make up the rest.

    The number of gingerbread cookies that he ate was two times the number of sugar cookies that he ate. What percentage of the cookies that he ate were gingerbread cookies?

    1. \(25\%\)
    2. \(28\%\)
    3. \(30\%\)
    4. \(35\%\)
    5. \(38\%\)

  7. If \(\dfrac{1}{6} + \dfrac{1}{3} = \dfrac{1}{x}\), the value of \(x\) is

    1. \(9\)
    2. \(6\)
    3. \(18\)
    4. \(2\)
    5. \(3\)

  8. Which of the following integers is equal to a perfect square?

    1. \(2^3\)
    2. \(3^5\)
    3. \(4^7\)
    4. \(5^9\)
    5. \(6^{11}\)

  9. The sum of five consecutive odd integers is \(125\). The smallest of these integers is

    1. \(17\)
    2. \(19\)
    3. \(21\)
    4. \(25\)
    5. \(29\)

  10. Two standard six-sided dice are rolled. What is the probability that the product of the two numbers rolled is \(12\)?

    1. \(\frac{3}{36}\)
    2. \(\frac{4}{36}\)
    3. \(\frac{6}{36}\)
    4. \(\frac{8}{36}\)
    5. \(\frac{9}{36}\)

Part B: Each correct answer is worth 6.

  1. Arturo has an equal number of \(\$5\) bills, of \(\$10\) bills, and of \(\$20\) bills. The total value of these bills is \(\$700\). How many \(\$5\) bills does Arturo have?

    1. \(16\)
    2. \(24\)
    3. \(12\)
    4. \(20\)
    5. \(28\)

  2. The Cayley Corner Store sells three types of toys: Exes, Wyes and Zeds. All Exes are identical, all Wyes are identical, and all Zeds are identical. The mass of \(2\) Exes equals the mass of \(29\) Wyes. The mass of \(1\) Zed equals the mass of \(16\) Exes. The mass of \(1\) Zed equals the mass of how many Wyes?

    1. \(3.625\)
    2. \(1.103\)
    3. \(232\)
    4. \(464\)
    5. \(928\)

  3. In the diagram, quadrilateral \(ABCD\) has \(AB = 20\), \(BC = 12\), and \(CD = 15\). Also, \(AB\) and \(CD\) are perpendicular to \(BC\).

    The perimeter of quadrilateral \(ABCD\) is

    1. \(47\)
    2. \(59\)
    3. \(84\)
    4. \(72\)
    5. \(60\)

  4. Ten numbers have an average (mean) of \(87\). Two of those numbers are \(51\) and \(99\). The average of the other eight numbers is

    1. \(90\)
    2. \(89\)
    3. \(88\)
    4. \(91\)
    5. \(92\)

  5. A rectangle has width \(x\) and length \(y\), as shown in Figure 1. The rectangle is cut along the horizontal and vertical dotted lines in Figure 1 to produce four smaller rectangles as shown in Figure 2.

    The vertical line starts at a point about one-third of the way along the top horizontal side of width x, and ends at the bottom side. The horizontal line starts at a point about one-third of the way down the left vertical side of length y, and ends at the right side. The four resulting rectangles are pulled apart.

    The sum of the perimeters of these four rectangles in Figure 2 is \(24\). The value of \(x + y\) is

    1. \(6\)
    2. \(8\)
    3. \(9.6\)
    4. \(12\)
    5. \(16\)

  6. Suppose that \(\sqrt{\dfrac{1}{2} \times \dfrac{2}{3} \times \dfrac{3}{4} \times \dfrac{4}{5} \times \cdots \times \dfrac{n-1}{n}} = \dfrac{1}{8}\). (The expression under the square root is the product of \(n-1\) fractions.) The value of \(n\) is

    1. \(81\)
    2. \(64\)
    3. \(16\)
    4. \(256\)
    5. \(100\)

  7. Each of the four digits of the integer \(2024\) is even. How many integers between \(1000\) and \(9999\), inclusive, have the property that all four of their digits are even?

    1. \(500\)
    2. \(625\)
    3. \(96\)
    4. \(54\)
    5. \(256\)

  8. The line with equation \(y = 3x + 5\) is translated \(2\) units to the right. The equation of the resulting line is

    1. \(y = 3x + 3\)
    2. \(y = 3x - 1\)
    3. \(y = 3x + 11\)
    4. \(y = 3x + 7\)
    5. \(y = 5x + 5\)

  9. In the diagram, \(\triangle ABC\) is right-angled at \(C\). Points \(D\), \(E\), \(F\) are on \(AB\), points \(G\), \(H\), \(J\) are on \(AC\), point \(K\) is on \(EH\), point \(L\) is on \(FJ\), and point \(M\) is on \(BC\) so that \(DKHG\), \(ELJH\) and \(FMCJ\) are squares.

    The area of \(DKHG\) is \(16\) and the area of \(ELJH\) is \(36\). The area of square \(FMCJ\) is

    1. \(64\)
    2. \(52\)
    3. \(100\)
    4. \(81\)
    5. \(75\)

  10. Jiwei and Hari entered a race. Hari finished the race in \(\frac{4}{5}\) of the time it took Jiwei to finish. The next time that they raced the same distance, Jiwei increased his average speed from the first race by \(x\)%, while Hari maintained the same average speed as in the first race. In this second race, Hari finished the race in the same amount of time that it took Jiwei to finish. The value of \(x\) is

    1. \(20\)
    2. \(25\)
    3. \(35\)
    4. \(40\)
    5. \(50\)

Part C: Each correct answer is worth 8.

Each correct answer is an integer from 0 to 99, inclusive.

  1. A \(3 \times 3\) table starts with every entry equal to 0 and is modified using the following steps:
    (i) adding 1 to all three numbers in any row;
    (ii) adding 2 to all three numbers in any column.

    After step (i) has been used a total of \(a\) times and step (ii) has been used a total of \(b\) times, the table appears as shown.

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

    What is the value of \(a+b\)?

  2. For how many integers \(m\) does the line with the equation \(y = mx\) intersect the line segment with endpoints \((20, 24)\) and \((4, 202)\)?

  3. Four semi-circles are arranged so that their diameters form a \(6\) by \(8\) rectangle. A circle is drawn through the four vertices of the rectangle. In the diagram, the region inside the four semi-circles but outside the circle is shaded. The total area of the shaded region is \(A\). What is the integer closest to \(A\)?

  4. A park has four paths, as shown in the map below.

    There are three locations on the map, labelled A, B, and C, connected by four paths. Each path has an arrow indicating a direction. There is a path from A to B. There is another path from B to A. There is a path from B to C. There is a path from B back to B.

    It takes \(2\) minutes to walk along the path from \(A\) to \(B\), \(3\) minutes to walk along the path from \(B\) to \(A\), \(3\) minutes to walk along the path from \(B\) to \(C\), and \(3\) minutes to walk around the path that begins and ends at \(B\). Rasheeqa goes for a walk, starting at \(A\), walking only in the directions indicated along the paths, never stopping to rest, and finishing at \(C\).

    If the walk takes a total of \(t\) minutes, how many possible values of \(t\) are there with \(t\leq 100\)?

  5. Erin has an empty \(1 \times 7\) grid consisting of \(1 \times 1\) squares:

    and follows the process below to construct a pattern:
    (i) Place an X in any empty square.
    (ii) If three or more consecutive squares each contain an X, stop and do not add any more X’s; otherwise, go to step (i) and continue the process.

    For example, in a smaller \(1 \times 4\) grid, there are \(3\) different patterns that can be constructed:

    The first pattern has Xs in the last three squares, the second has Xs in the first three squares, and the third has Xs in all four squares.

    (The last pattern may be obtained by placing X’s, in order, in squares 1, 2, 4, and then 3.) By applying this process starting with the empty \(1 \times 7\) grid, how many different possible patterns can Erin construct?

Further Information

For students...

Thank you for writing the Cayley Contest!

Encourage your teacher to register you for 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