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 value of is
The average (mean) of two numbers is 7. One of the numbers is 5.
The other number is
Gauravi walks every day. One Monday, she walks 500 m. On each day
that follows, she increases her distance by 500 m from the previous day.
On what day of the week will she walk exactly 4500 m?
What is the largest number of squares with side length 2 that can
be arranged, without overlapping, inside a square with side length
8?
One integer is selected at random from the following list of 15
integers: The
probability that the selected integer is equal to is . What is the value of ?
In the diagram, points , and form a triangle.
The area of is
The expression is
equal to
If and for some positive integers
and , the value of is
It takes Pearl 7 days to dig 4 holes. It takes Miguel 3 days to
dig 2 holes. If they work together and each continues digging at these
same rates, how many holes in total will they dig in 21 days?
If for some positive integers and , then the value of is
Part B: Each correct
answer is worth 6.
Dhruv is older than Bev. Bev is older than Elcim. Elcim is
younger than Andy. Andy is younger than Bev. Bev is younger than Cao.
Who is the third oldest?
Suppose that is an odd
integer and is an even integer.
How many of the following expressions are equal to an odd integer?
Seven identical rectangles are used to create two larger
rectangles, as shown in Figure A and Figure B.
Figure A
Figure B
The ratio of the perimeter of Figure A to the perimeter of Figure B
is
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?
A positive integer is
input into a machine. If is odd,
the output is . If is even, the output is . This process can be repeated using
each successive output as the next input. For example, if the input is
and the machine is used three
times, the final output is 12. If the input is and the machine is used 51 times,
the final output is
The remainder when 111 is divided by 10 is 1. The remainder when
111 is divided by the positive integer is 6. The number of possible values of
is
An aluminum can in the shape of a cylinder is closed at both
ends. Its surface area is . If the radius of the can were doubled, its surface area
would be . If
instead the height of the can were doubled, what would its surface area
be?
(The surface area of a cylinder with radius and height is equal to .)
Aria and Bianca walk at different, but constant speeds. They each
begin at 8:00 a.m. from the opposite ends of a road and walk directly
toward the other’s starting point. They pass each other at 8:42 a.m.
Aria arrives at Bianca’s starting point at 9:10 a.m. Bianca arrives at
Aria’s starting point at
In the diagram, is right-angled at ,
, and . Also, is the midpoint of and is the point on so that is perpendicular to .
The area of is
A sequence of numbers has its terms defined by for every
integer . For example,
.
What is the largest positive integer for which the sum of the first terms (that is, ) is
less than 1.499?
Part C: Each correct
answer is worth 8.
Each correct answer is an integer from 0 to 99, inclusive.
Gustave has 15 steel bars of masses 1 kg, 2 kg, 3 kg, , 14 kg, 15 kg. He also has 3 bags
labelled , , . He places two steel bars in each bag
so that the total mass in each bag is equal to kg. How many different values of are possible?
A rectangle with dimensions 100 cm by 150 cm is tilted so that
one corner is 20 cm above a horizontal line, as shown.
To the nearest centimetre, the height of vertex above the horizontal line is cm. What is the value of ?
For how many positive integers do the lines with equations and intersect at a point whose
coordinates are positive integers?
There are functions
with the following properties:
for
some integers , and with , and
and for some prime numbers and with .
For each such function, the value of is calculated. The sum of all
possible values of is . What are the rightmost two digits of
?
In the grid
shown, the central square contains the integer 5. The remaining eight
squares contain , , , , , , , , which are each to be replaced with an
integer from 1 to 9, inclusive. Integers can be repeated. There are ways to complete the grid so that the
sums of the integers along each row, along each column, and along the
two main diagonals are all divisible by 5.
What are the rightmost two digits of ?
Further Information
For students...
Thank you for writing the Fermat Contest!
Encourage your teacher to register you for the Hypatia Contest which will be written in April.