2016 Fermat Contest
(Grade 11)
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.
If , and , the value of is
A cube has 12 edges, as shown.

How many edges does a square-based pyramid have?
The expression equals
An oblong number is the number of dots in a rectangular grid with one more row than column. The first four oblong numbers are 2, 6, 12, and 20, and are represented below:
What is the 7th oblong number?
In the diagram, point is the midpoint of .

The coordinates of are
Carrie sends five text messages to her brother each Saturday and five text messages to her brother each Sunday. Carrie sends two text messages to her brother on each of the other days of the week. Over the course of four full weeks, how many text messages does Carrie send to her brother?
The value of is
If , the value of is
If % of 60 is 12, then 15% of is
In the diagram, square has side length 2. Points , , , and are the midpoints of the sides of .

What is the ratio of the area of square to the area of square ?
Part B: Each correct answer is worth 6.
In the diagram, is right-angled at and .

If point is on so that and , the perimeter of is
How many of the positive divisors of 128 are perfect squares larger than 1?
The numbers are three consecutive terms in an arithmetic sequence. What is the value of ?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, are the first four terms of an arithmetic sequence.)
Suppose that and are integers with . If the average (mean) of the numbers is 13, then the number of possible pairs is
Hicham runs 16 km in 1.5 hours. He runs the first 10 km at an average speed of 12 km/h. What is his average speed for the last 6 km?
If is one of the solutions of the equation , the other solution of this equation is
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
Suppose that and satisfy and .
The value of is
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
In the diagram, represents a rectangular piece of paper. The paper is folded along a line so that . When the folded paper is flattened, points and have moved to points and , respectively, and crosses at .

The measure of is
Part C: Each correct answer is worth 8.
Box 1 contains one gold marble and one black marble. Box 2 contains one gold marble and two black marbles. Box 3 contains one gold marble and three black marbles. Whenever a marble is chosen randomly from one of the boxes, each marble in that box is equally likely to be chosen. A marble is randomly chosen from Box 1 and placed in Box 2. Then a marble is randomly chosen from Box 2 and placed in Box 3. Finally, a marble is randomly chosen from Box 3. What is the probability that the marble chosen from Box 3 is gold?
If and are real numbers, the minimum possible value of the expression is
Seven coins of three different sizes are placed flat on a table, arranged as shown in the diagram. Each coin, except the centre one, touches three other coins. The centre coin touches all of the other coins.

If the coins labelled have a radius of 3 cm, and those labelled have radius 2 cm, then the radius of the coin labelled is closest to
For any real number , denotes the largest integer less than or equal to .For example, and .
If is the sum of all integers with and for which is divisible by , then equals
The set contains 2045 elements. A subset of is called triple-free if no element of equals three times another element of . For example, is triple-free, but is not triple-free. The triple-free subsets of that contain the largest number of elements contain exactly elements. There are triple-free subsets of that contain exactly elements. The integer can be written in the form , where and are distinct prime numbers and and are positive integers. If , then the last three digits of are
Further Information
For students...
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