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.
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.
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").
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 integer 119 is a multiple of
Which of the following fractions has the greatest value?
The pattern of shapes is repeated to form
the sequence The 22nd shape in the sequence
is
The expression is equal to
Yihana walks for 10 minutes. A graph of her elevation in metres
versus time in minutes is shown.
A broken line graph entitled Yihana’s Walk. The horizontal axis is
Time in minutes and the vertical axis is Elevation in metres. The five line segments in the graph are as follows:
to ,
to ,
to ,
to , and
to .
The length of time for which she was walking uphill is
Points , , , , , and are evenly spaced around the circle
with centre , as shown.
The measure of is
A rectangle has positive integer side lengths and an area of 24.
The perimeter of the rectangle cannot be
The operation is
defined by for all integers and with . For example, . If , what is the value of ?
If is of and is of , then what percentage is of ?
Part B: Each correct
answer is worth 6.
A store sells jellybeans at a fixed price per gram. The price for
250 g of jellybeans is $7.50. What mass of jellybeans sells for
$1.80?
An equilateral triangle is made of cardboard and lies on a table.
Paola stands in front of the table and sees the triangle in the position
shown.
She flips the triangle over, keeping edge on the table throughout the flip. From
this position, Paola then flips the triangle again, this time keeping
edge on the table throughout the
flip. What is the resulting position of the triangle that Paola
sees?
Two identical smaller cubes are stacked next to a larger cube.
Each of the two smaller cubes has a volume of 8. The combined height of
the smaller cubes equals the height of the larger cube. What is the
volume of the larger cube?
The integer 48 178 includes the block of digits 178. The three
integers 51 870, 19 728 and 38 717 do not include the block of digits
178. How many integers between 10 000 and 100 000 include the block of
digits 178?
The integers , and satisfy the equations and and . The value of is
In the diagram, hexagon has interior right angles at , , , , and and an exterior right angle at .
Also, , , and . The perimeter of is closest to
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a
drawer. Without looking, he randomly pulls shirts from his drawer one at
a time. He would like a set of shirts that includes either 3 of the same
colour or 3 of different colours. What is the minimum number of shirts
that Zebadiah has to pull out to guarantee that he has such a
set?
At the beginning of the first day, a box contains 1 black ball, 1
gold ball, and no other balls. At the end of each day, for each gold
ball in the box, 2 black balls and 1 gold ball are added to the box;
this means that at the end of the first day, there are 5 balls in the
box. If no balls are removed from the box, how many balls are in the box
at the end of the seventh day?
The area of the triangular region bounded by the -axis, the -axis and the line with equation is one-quarter of the area of
the triangular region bounded by the -axis, the line with equation and the line with equation
, where . What is the value of ?
If and are positive integers that satisfy the
equation , the smallest
possible value for is
Part C: Each correct
answer is worth 8.
Each correct answer is an integer from 0 to 99, inclusive.
There are exactly four ordered pairs of positive integers that satisfy the equation . Mehdi writes down the four
values of and adds the smallest
and largest of these values. What is this sum?
In the diagram, two circles are centred at . The smaller circle has a radius of 1
and the larger circle has a radius of 3. Points and are placed on the larger circle so that
the areas of the two shaded regions are equal.
If , what is
the value of ?
Andreas, Boyu, Callista, and Diane each randomly choose an
integer from 1 to 9, inclusive. Each of their choices is independent of
the other integers chosen and the same integer can be chosen by more
than one person. The probability that the sum of their four integers is
even is equal to
for some positive integer . What
is the sum of the squares of the digits of ?
A cube with edge length is
balanced on one of its vertices on a horizontal table such that the
diagonal from this vertex through the interior of the cube to the
farthest vertex is vertical. When the sun is directly above the top
vertex, the shadow of the cube on the table is a regular hexagon. The
area of this shadow can be written in the form , where and are positive integers and is not divisible by any perfect square
larger than 1. What is the value of ?
There are tokens arranged
in a circle for some positive integer . Moving clockwise around the circle,
the tokens are labelled, in order, with the integers from 1 to . Starting from the token labelled 1,
Évariste:
Removes the token at the current position.
Moves clockwise to the next remaining token.
Moves clockwise again to the next remaining token.
Repeats steps (i) to (iii) until only one token remains.
When , the number on the
last remaining token is . There
are other integers for which the
number on the last remaining token is also . What are the rightmost two digits of
the smallest possible value of ?
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.