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2020 Cayley Contest
(Grade 10)

Tuesday, February 25, 2020
(in North America and South America)

Wednesday, February 26, 2020
(outside of North American and South America)

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©2020 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 \(\dfrac{20-20}{20+20}\) is

    1. \(0\)
    2. \(1\)
    3. \(10\)
    4. \(-2\)
    5. \(2\)

  2. When \(x=3\) and \(y=4\), the value of \(xy-x\) is

    1. \(3\)
    2. \(4\)
    3. \(12\)
    4. \(9\)
    5. \(15\)

  3. The points \(O(0, 0)\), \(P(0,3)\), \(Q\), and \(R(5,0)\) form a
    rectangle, as shown.

    The coordinates of \(Q\) are

    1. \((5,5)\)
    2. \((5,3)\)
    3. \((3,3)\)
    4. \((2.5,1.5)\)
    5. \((0,5)\)

  4. Which of the following numbers is less than \(\frac{1}{20}\)?

    1. \(\frac{1}{15}\)
    2. \(\frac{1}{25}\)
    3. \(0.5\)
    4. \(0.055\)
    5. \(\frac{1}{10}\)

  5. In the diagram, point \(Q\) lies on \(PR\) and point \(S\) lies on \(QT\).

    In triangle SPQ, angle SPQ is x degrees, Side QS is extended to point T (T is closer to S), and angle TSP is 50 degrees. Side PQ is extended to point R (R closer to Q), and angle SQR is 150 degrees.

    What is the value of \(x\)?

    1. \(10\)
    2. \(30\)
    3. \(50\)
    4. \(40\)
    5. \(20\)

  6. Matilda counted the birds that visited her bird feeder yesterday. She summarized the data in the bar graph shown.

    The vertical axis of a bar graph is labelled 'Number' and the horizontal axis is labelled 'Type of Bird'. There are three bars on the graph: Goldfinch has a height of 6, Sparrow has a height of 9 and Grackle has a height of 5.

    The percentage of birds that were goldfinches is

    1. \(15\%\)
    2. \(20\%\)
    3. \(30\%\)
    4. \(45\%\)
    5. \(60\%\)

  7. The average of the two positive integers \(m\) and \(n\) is 5. What is the largest possible value for \(n\)?

    1. \(5\)
    2. \(7\)
    3. \(9\)
    4. \(11\)
    5. \(13\)

  8. Roman wins a contest with a prize of $200. He gives 30% of the prize to Jackie. He then splits 15% of what remains equally between Dale and Natalia. How much money does Roman give Dale?

    1. \(\$10.50\)
    2. \(\$15.00\)
    3. \(\$4.50\)
    4. \(\$25.50\)
    5. \(\$59.50\)

  9. Shaded and unshaded squares are arranged in rows so that:

    The first 4 rows are shown.

    The first row has one unshaded square. The second row has three squares with the middle square shaded. The third row has five squares with the second and fourth square shaded. The fourth row has seven squares with the second, fourth, and sixth square shaded.

    The number of shaded squares in the 2020th row is

    1. \(2022\)
    2. \(2021\)
    3. \(2020\)
    4. \(2019\)
    5. \(2018\)

  10. In the diagram, pentagon \(PQRST\) has \(PQ = 13\), \(QR =18\), \(ST=30\), and a perimeter of 82. Also, \(\angle QRS = \angle RST = \angle STP = 90^\circ\).

    Side PT is longer than side RS.

    The area of the pentagon \(PQRST\) is

    1. \(306\)
    2. \(297\)
    3. \(288\)
    4. \(279\)
    5. \(270\)

Part B: Each correct answer is worth 6.

  1. The sum of the first 9 positive integers is 45; in other words, \[1+2+3+4+5+6+7+8+9=45\] What is the sum of the first 9 positive multiples of 5? In other words, what is the value of \(5+10+15+\cdots + 40+45\)?

    1. \(225\)
    2. \(250\)
    3. \(180\)
    4. \(150\)
    5. \(450\)

  2. The volume of a rectangular prism is 21. Its length, width and height are all different positive integers. The sum of its length, width and height is

    1. \(11\)
    2. \(13\)
    3. \(15\)
    4. \(9\)
    5. \(17\)

  3. If \(2^n = 8^{20}\), what is the value of \(n\)?

    1. \(10\)
    2. \(60\)
    3. \(40\)
    4. \(16\)
    5. \(17\)

  4. Juliana chooses three different numbers from the set \(\{-6,-4,-2,0,1,3,5,7\}\) and multiplies them together to obtain the integer \(n\). What is the greatest possible value of \(n\)?

    1. \(168\)
    2. \(0\)
    3. \(15\)
    4. \(105\)
    5. \(210\)

  5. A bag contains only green, yellow and red marbles. The ratio of green marbles to yellow marbles to red marbles in the bag is \(3 : 4 : 2\). If 63 of the marbles in the bag are not red, the number of red marbles in the bag is

    1. \(14\)
    2. \(18\)
    3. \(27\)
    4. \(36\)
    5. \(81\)

  6. In the diagram, the circle has centre \(O\) and square \(OPQR\) has vertex \(Q\) on the circle.

    Points R and P lie inside the circle, so that the square OPQR is contained within the circle.

    If the area of the circle is 72\(\pi\), the area of the square is

    1. \(38\)
    2. \(48\)
    3. \(25\)
    4. \(12\)
    5. \(36\)

  7. Carley made treat bags. Each bag contained exactly 1 chocolate, 1 mint, and 1 caramel. The chocolates came in boxes of 50. The mints came in boxes of 40. The caramels came in boxes of 25. Carley made no incomplete treat bags and there were no unused chocolates, mints or caramels. What is the minimum total number of boxes that Carley could have bought?

    1. \(19\)
    2. \(17\)
    3. \(44\)
    4. \(25\)
    5. \(9\)

  8. Nate is driving to see his grandmother. If he drives at a constant speed of 40 km/h, he will arrive 1 hour late. If he drives at a constant speed of 60 km/h, he will arrive 1 hour early. At what constant speed should he drive to arrive just in time?

    1. \(56\mbox{ km/h}\)
    2. \(80\mbox{ km/h}\)
    3. \(54\mbox{ km/h}\)
    4. \(48\mbox{ km/h}\)
    5. \(58\mbox{ km/h}\)

  9. A multiple choice test has 10 questions on it. Each question answered correctly is worth 5 points, each unanswered question is worth 1 point, and each question answered incorrectly is worth 0 points. How many of the integers between 30 and 50, inclusive, are not possible total scores?

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

  10. For how many pairs \((m,n)\) with \(m\) and \(n\) integers satisfying \(1 \leq m \leq 100\) and \(101 \leq n \leq 205\) is \(3^m+7^n\) divisible by 10?

    1. \(2600\)
    2. \(2626\)
    3. \(2601\)
    4. \(2650\)
    5. \(2625\)

Part C: Each correct answer is worth 8.

  1. How many points \((x,y)\), with \(x\) and \(y\) both integers, are on the line with equation \(y = 4x + 3\) and inside the region bounded by \(x=25\), \(x=75\), \(y=120\), and \(y=250\)?

    1. \(44\)
    2. \(36\)
    3. \(40\)
    4. \(32\)
    5. \(48\)

  2. In the diagram, points \(S\) and \(T\) are on sides \(QR\) and \(PQ\), respectively, of \(\triangle PQR\) so that \(PS\) is perpendicular to \(QR\) and \(RT\) is perpendicular to \(PQ\).

    If \(PT=1\), \(TQ=4\), and \(QS=3\), what is the length of \(SR\)?

    1. \(3\)
    2. \(\frac{11}{3}\)
    3. \(\frac{15}{4}\)
    4. \(\frac{7}{2}\)
    5. \(4\)

  3. Ricardo wants to arrange three 1s, three 2s, two 3s, and one 4 to form nine-digit positive integers with the properties that

    (For example, the integer \(121\,321\,234\) satisfies these properties.) In total, how many such nine-digit positive integers can Ricardo make?

    1. \(278\)
    2. \(260\)
    3. \(254\)
    4. \(272\)
    5. \(266\)

  4. A cube with vertices \(FGHJKLMN\) has edge length 200. Point \(P\) is on \(HG\), as shown. The shortest distance from \(G\) to a point inside \(\triangle PFM\) is 100.

    The vertices of the cube are labelled as follows: the top face is KNHJ and the bottom face is LMGF so that L is directly below K, M is directly below N, G is directly below H, and F is directly below J. Point P lies on the vertical edge HG.

    Which of the following is closest to the length of \(HP\)?

    1. \(53\)
    2. \(55\)
    3. \(57\)
    4. \(59\)
    5. \(61\)

  5. How many positive integers \(n \leq 20\,000\) have the properties that \(2n\) has 64 positive divisors including 1 and \(2n\), and \(5n\) has 60 positive divisors including 1 and \(5n\)?

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

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