2019 Fermat Contest
(Grade 11)
Tuesday, February 26, 2019
(in North America and South America)
Wednesday, February 27, 2019
(outside of North American and South America)

©2010 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.
What is the remainder when 14 is divided by 5?
Which of the following is equal to for all values of and ?
The value of is
In the diagram, point is on the number line at 3 and is at 33. The number line between and is divided into six equal parts by the points .
What is the sum of the lengths of and ?
Mike rides his bicycle at a constant speed of 30 km/h. How many kilometres does Mike travel in 20 minutes?
In the diagram, is a rectangle. Also, , and are congruent.
What fraction of the area of rectangle is shaded?
The town of Cans is north of the town of Ernie. The town of Dundee is south of Cans but north of Ernie. The town of Arva is south of the town of Blythe and is north of both Dundee and Cans. The town that is the most north is
The product is divisible by . The largest possible integer value of is
The average of and is
The digits , , , , and can be used, each exactly once, to form many five-digit integers. Of these integers, is the one that is as close as possible to 30 000. What is the tens digit of ?
Part B: Each correct answer is worth 6.
Line is perpendicular to the line with equation . Line has the same -intercept as the line with equation . The -intercept of line is
The first part of the Genius Quiz has 30 questions and the second part has 50 questions. Alberto answered exactly 70% of the 30 questions in the first part correctly. He answered exactly 40% of the 50 questions in the second part correctly. The percentage of all of the questions on the quiz that Alberto answered correctly is closest to
Tanis looked at her watch and noticed that, at that moment, it was minutes after 7:00 a.m. and minutes before 8:00 a.m. for some value of . What time was it at that moment?
The letters A, B, C, D, and E are to be placed in the grid so that each of these letters appears exactly once in each row and exactly once in each column.
Which letter will go in the square marked with ?
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. How many different-looking arrangements are possible?
In the diagram, each line segment has length or . Also, each pair of adjacent sides is perpendicular.
If the area of the figure is 252 and , the perimeter of the figure is
The five sides of a regular pentagon are all equal in length. Also, all interior angles of a regular pentagon have the same measure. In the diagram, is a regular pentagon and is equilateral.

The measure of obtuse is
How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6?
The function has the properties that and for every integer . What is the value of ?
The vertices of an equilateral triangle lie on a circle with radius 2. The area of the triangle is
Part C: Each correct answer is worth 8.
In the multiplication shown, each of , , , , and is a digit.

The value of is
In the diagram, two circles touch at . Also, and are perpendicular diameters of the larger circle that intersect at . Point is on and is a diameter of the smaller circle. The smaller circle intersects at , as shown.

If and , what is the sum of the lengths of the diameters of the two circles?
How many positive integers with can be expressed as the sum of four or more consecutive positive integers?
Consider the quadratic equation where is a real number. This equation has two distinct real solutions which are both negative exactly when , for some real numbers and . The value of is
In , point is on with . Suppose that and for some integers and with and for which is a multiple of .

Suppose also that the perimeter of is and that the number of possible integer values for is . The value of is
Further Information
For students...
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