2021 Cayley Contest
(Grade 10)
Thursday, February 23, 2021
(in North America and South America)
Wednesday, February 24, 2021
(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.
- Do not open the Contest booklet until you are told to do so.
- You may use rulers, compasses and paper for rough work.
- 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.
- On your response form, print your school name and city/town in the box in the upper right corner.
- 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.
- 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.
- Scoring:
- Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
- There is no penalty for an incorrect answer.
- Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
- Diagrams are not drawn to scale. They are intended as aids only.
- When your supervisor tells you to begin, you will have sixty minutes of working time.
- 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:
- There is no penalty for an incorrect answer.
- Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
Part A: Each correct answer is worth 5.
The expression is equal to
The ones (units) digit of 542 is 2. When 542 is multiplied by 3, the ones (units) digit of the result is
Some of the squares in a grid are shaded, as shown.

What is the perimeter of the shaded region?
If , the value of is
Which of the following is equal to 110% of 500?
Eugene swam on Sunday, Monday and Tuesday. On Monday, he swam for 30 minutes. On Tuesday, he swam for 45 minutes. His average swim time over the three days was 34 minutes. For how many minutes did he swim on Sunday?
For which of the following values of is ?
A square piece of paper has a dot in its top right corner and is lying on a table. The square is folded along its diagonal, then rotated clockwise about its centre, and then finally unfolded, as shown.
The resulting figure is





In 12 years, Janice will be 8 times as old as she was 2 years ago. How old is Janice now?
In the diagram, pentagon is constructed from equilateral and square .
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The measure of is equal to
Part B: Each correct answer is worth 6.
In the diagram, which of the following points is at a different distance from than the rest of the points?

If and and , then equals
In the diagram, is a straight line segment.

If , what is the value of ?
The line with equation is reflected in the -axis. What is the -intercept of the resulting line?
Amy bought and then sold avocados, for some positive integer . She made a profit of $100. (Her profit is the difference between the total amount that she earned by selling the avocados and the total amount that she spent in buying the avocados.) She paid $2 for every 3 avocados. She sold every 5 avocados for $4. What is the value of ?
If , the value of is
A group of friends are sharing a bag of candy.
On the first day, they eat of the candies in the bag.
On the second day, they eat of the remaining candies.
On the third day, they eat of the remaining candies.
On the fourth day, they eat of the remaining candies.
On the fifth day, they eat of the remaining candies.
At the end of the fifth day, there is 1 candy remaining in the bag.
How many candies were in the bag before the first day?
Elina and Gustavo leave Cayley H.S. at 3:00 p.m. Elina runs north at a constant speed of 12 km/h. Gustavo walks east at a constant speed of 5 km/h. After 12 minutes, Elina and Gustavo change direction and travel directly towards each other, still at 12 km/h and 5 km/h, respectively. The time that they will meet again is closest to
In the diagram, eight circles, each of radius 1, are drawn inside a rectangle.

Four of the circles are tangent to two sides of the rectangle and to two other circles. Four of the circles are tangent to one side of the rectangle and to three other circles. A region has been shaded, as shown.
It consists of three spaces (each space bounded by a different set of four circles), as well as four of the circles themselves. The area of this region is closest to
How many four-digit positive integers are divisible by both 12 and 20, but are not divisible by 16?
Part C: Each correct answer is worth 8.
The variables , , , , , and represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that The value of is
The cells of a grid are to be filled with integers so that the average value of the entries along each row, each column, and each diagonal is the same. The integers 10, 64 and 70 are entered, as shown.

When the remaining six squares are filled in to complete the grid, what integer replaces ?
A special six-sided die has its faces numbered 1 through 6 and has the property that rolling each number is times as likely as rolling a . For example, the probability of rolling a 5 is 5 times the probability of rolling a 1, while the probability of rolling a 2 is 2 times the probability of rolling a 1. Robbie and Francine play a game where they each roll this die three times, and the total of their three rolls is their score. The winner is the player with the highest score; if the two players are tied, neither player wins. After two rolls each, Robbie has a score of 8 and Francine has a score of 10. The probability that Robbie will win can be written in lowest terms as , where and are positive integers. What is value of ?
In the diagram, is a diameter of the circular base of the cylinder. is a diameter of the top face of the cylinder and is directly above , as shown.
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Point is on the circumference of the top face, halfway between and . Point is on the cylinder and is directly above . Suppose that and , where and are integers with . If and , what is the remainder when the integer equal to is divided by 100?
The points , and form a triangle whose area is less than or equal to 10. Let be the region formed by all such points with and . When written as a fraction in lowest terms, the area of is equal to for some positive integers and . The value of is
Further Information
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