2016 Cayley Contest
(Grade 10)
Wednesday, February 24, 2016
(in North America and South America)
Thursday, February 25, 2016
(outside of North American and South America)

©2015 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 value of is
-
Maya asked the 20 math teachers at her school to tell her their favourite shape. She represented their answers
on the bar graph shown.

The number of teachers who did not pick “Square" as their favourite shape was
-
The expression is equal to
-
If each of Bill’s steps is metre long, how many steps does
Bill take to walk 12 metres in a straight line?
-
In the diagram, is on .

The value of is
-
If the line that passes through the points and has a slope of 2, the value of
is
-
A soccer team played three games. Each game ended in a win, loss, or tie. (If a game finishes with both teams
having scored the same number of goals, the game ends in a tie.) In total, the team scored more goals than were
scored against them. Which of the following combinations of outcomes is not possible for this team?
-
The first five letters of the alphabet are assigned the values , , , , and . The value of a word equals
the sum of the values of its letters. For example, the value of is
. Which of the following words has the largest value?
-
Grace writes a sequence of 20 numbers. The first number is 43 and each number after the first is 4 less than
the number before it, so her sequence starts . How many
of the numbers that Grace writes are positive?
-
Five students play chess matches against each other. Each student plays three matches against each of the other
students. How many matches are played in total?
Part B: Each correct answer is worth 6.
-
In the diagram, is perpendicular to , is perpendicular to , and is perpendicular to .

If , , , and , then the distance from to is
-
Alejandro has a box that contains 30 balls, numbered from 1 to 30. He randomly selects a ball from the box
where each ball is equally likely to be chosen. Which of the following is most likely?
-
Which of the following fractions is both larger than and
smaller than ?
-
The number of zeros in the integer equal to is
-
What is the tens digit of the smallest positive integer that is divisible by each of 20, 16 and 2016?
-
The triangle shown is reflected in the -axis and the resulting triangle
is reflected in the -axis.

Which of the following best represents the final position of the triangle?





-
In the diagram, the perimeter of square is and the perimeter of is
.

Which of the following expressions in terms of is equal to the perimeter of
pentagon ?
-
When three positive integers are added in pairs, the resulting sums are 998, 1050 and 1234. What is the
difference between the largest and smallest of the three original positive integers?
-
A total of points are equally spaced around a circle and are labelled
with the integers 1 to , in order. Two points are called diametrically
opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35
are diametrically opposite, then equals
-
There are students in the math club at Scoins Secondary School. When
Mrs. Fryer tries to put the students in groups of 4, there is one group
with fewer than 4 students, but all of the other groups are complete. When she tries to put the students in groups of 3, there are 3 more complete groups than there were
with groups of 4, and there is again exactly one group that is not complete. When she tries to put the students in groups of 2, there are 5 more complete groups than there were
with groups of 3, and there is again exactly one group that is not complete. The sum of the digits of the
integer equal to is
Part C: Each correct answer is worth 8.
-
In her last basketball game, Jackie scored 36 points. These points raised the average (mean) number of points
that she scored per game from 20 to 21. To raise this average to 22 points, how many points must Jackie score in
her next game?
-
Alain and Louise are driving on a circular track with radius 25 km. Alain leaves the starting line first, going
clockwise around the track at a speed of 80 km/h. Fifteen minutes after Alain starts, Louise leaves the same
starting line, going counter-clockwise around the track at a speed of 100 km/h. For how many hours will Louise
have been driving when the two of them pass each other for the fourth time?
-
Suppose that is a regular octagon. (A regular octagon is
an octagon with eight equal side lengths and eight equal interior angles.) There are 70 ways in which four of
its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended
infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains
the octagon?
-
What is the sum of all numbers which can be written in the form where and are positive integers with and
for which there are exactly 19 integers that satisfy ?
-
A new language uses only the letters A, B, C, D, and E. The letters A and E are called vowels, while
the letters B, C and D are called consonants. A sequence of letters is called a word if it
does not include the same letter twice in a row, and it does not include two vowels in a row. How many words are
there in this language that are 10 letters long and that begin with a vowel?
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
- Free copies of past contests
- Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
- Information about careers in and applications of mathematics and computer science
For teachers...
Visit our website cemc.uwaterloo.ca to
- Obtain information about future contests
- Look at our free online courseware for high school students
- Learn about our face-to-face workshops and our web resources
- Subscribe to our free Problem of the Week
- Investigate our online Master of Mathematics for Teachers
- Find your school's contest results