CEMC Banner

2026 Fermat Contest
(Grade 11)

Wednesday, February 25, 2026
(in North America and South America)

Thursday, February 26, 2026
(outside of North American and South America)

University of Waterloo Logo

©2026 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.

  1. Do not open the Contest booklet until you are told to do so.
  2. You may use rulers, compasses and paper for rough work.
  3. 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.
  4. On your response form, print your school name and city/town in the box in the upper right corner.
  5. 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.
  6. 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.
  7. 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").
  8. Scoring:
    1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
    2. There is no penalty for an incorrect answer.
    3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
  9. Diagrams are not drawn to scale. They are intended as aids only.
  10. When your supervisor tells you to begin, you will have sixty minutes of working time.
  11. 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:
  1. There is no penalty for an incorrect answer.
  2. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth 5.

  1. The cost to buy \(2\) F-MAT calculators is \(\$30\). The cost to buy \(4\) F-MAT calculators is

    1. \(\$15\)
    2. \(\$45\)
    3. \(\$60\)
    4. \(\$75\)
    5. \(\$120\)

  2. The value of \(\dfrac{101^2-101}{100}\) is

    1. \(101\)
    2. \(1\)
    3. \(100\)
    4. \(0\)
    5. \(10\,200\)

  3. A math club had \(24\) members in its first year. In the second year, the membership increased by \(50\%\). How many members were there in the second year?

    1. \(54\)
    2. \(48\)
    3. \(36\)
    4. \(42\)
    5. \(60\)

  4. The squares shown are each filled with a number so that the sum of the numbers in every group of three consecutive squares is \(24\). What number appears in the square marked with a question mark?

    There are seven squares in a row. Moving from left to right in the row, the first three squares have numbers 3, 5, and 16 and the last square has a question mark.

    1. \(3\)
    2. \(5\)
    3. \(8\)
    4. \(16\)
    5. \(24\)

  5. The right side of the equation below represents some number of \(4\)s added together. \[4^3 = 4 + 4 + 4 + \cdots + 4\]How many \(4\)s are in the sum?

    1. \(12\)
    2. \(24\)
    3. \(8\)
    4. \(16\)
    5. \(3\)

  6. Fermat buys \(8\) bananas and \(4\) apples. The cost of each banana is half the cost of each apple. If the total cost of the fruit is \(\$16.00\), what is the cost of each apple?

    1. \(\$0.50\)
    2. \(\$1.33\)
    3. \(\$4.00\)
    4. \(\$2.67\)
    5. \(\$2.00\)

  7. Leah went for a walk and recorded the number of robins, cardinals and blue jays that she saw in the bar graph shown. Unfortunately, she forgot to include numbers on the vertical axis. If a total of \(30\) robins, cardinals and blue jays were observed, how many robins did she see?

    The vertical axis of the bar graph starts at 0 and then has six equally spaced tick marks moving up the axis. The bar for Robins reaches the sixth tick mark. The bar for Cardinals reaches the third tick mark. The bar for Blue Jays reaches the first tick mark.

    1. \(9\)
    2. \(12\)
    3. \(15\)
    4. \(18\)
    5. \(21\)

  8. Suppose that \(a\), \(b\), \(c\), \(d\) are four consecutive positive integers with \(a<b<c<d\). The value of \((a+d)-(b+c)\) is

    1. \(-1\)
    2. \(0\)
    3. \(1\)
    4. \(2\)
    5. \(3\)

  9. A square has an area of \(10\). Each of its side lengths is multiplied by the same positive integer to produce a new square. Which of the following could be the area of the new square?

    1. \(60\)
    2. \(70\)
    3. \(80\)
    4. \(90\)
    5. \(100\)

  10. The faces of a six-sided die are numbered \(1\) to \(6\). The faces of an eight-sided die are numbered \(1\) to \(8\). Each die is rolled once. What is the probability that the same number is rolled on both dice?

    1. \(\dfrac{1}{64}\)
    2. \(\dfrac{1}{6}\)
    3. \(\dfrac{1}{48}\)
    4. \(\dfrac{1}{36}\)
    5. \(\dfrac{1}{8}\)

Part B: Each correct answer is worth 6.

  1. Maria drove \(500\) km in \(8\) hours. Andrea flew a helicopter \(1500\) km and averaged a speed that was four times Maria's average driving speed. How long did Andrea's flight take?

    1. \(1.5\) hours
    2. \(2\) hours
    3. \(4\) hours
    4. \(6\) hours
    5. \(8\) hours

  2. How many positive integers less than \(100\) can be written as the sum of three consecutive positive integers?

    1. \(97\)
    2. \(98\)
    3. \(94\)
    4. \(32\)
    5. \(33\)

  3. Points \(E\) and \(F\) are positioned on the sides of rectangle \(ABCD\) so that \(BF = ED = DC = AB= 8\) and \(EF = 10\), as shown.

    Point E is on side AD and point F is on side BC which is parallel to side AD.

    The area of rectangle \(ABCD\) is

    1. \(192\)
    2. \(176\)
    3. \(144\)
    4. \(208\)
    5. \(128\)

  4. Ava painted one picture each day for \(31\) consecutive days and numbered the paintings consecutively from \(1\) to \(31\). Three of the pictures that she painted on Sundays have even numbers. On what day did she paint the picture numbered 25?

    1. Monday
    2. Tuesday
    3. Wednesday
    4. Thursday
    5. Friday

  5. How many ordered pairs of positive integers \((x, y)\) have the property that the ratio \(x : 4\) equals the ratio \(9 : y\)?

    1. \(6\)
    2. \(7\)
    3. \(8\)
    4. \(9\)
    5. \(10\)

  6. In the diagram, point \(P\) lies inside square \(WXYZ\) so that \(\triangle PXY\) is an equilateral triangle.

    The measure of \(\angle WPZ\) is

    1. \(150\degree\)
    2. \(135\degree\)
    3. \(90\degree\)
    4. \(120\degree\)
    5. \(160\degree\)

  7. Each of the three nets shown is folded to create a cube.

    A net with a middle row of three squares containing numbers 2, 3, and 6 from left to right. Above the square with 6 is a square with 4. Below the square with 6 is a square with 5. To the right of the square with 5 is a square with 3.       A net with a top row of two squares containing numbers 2 and 4 from left to right. Below the 4 is a square with 6 and to the right of 6 there is a square with 8. Below the 8 is a square with 10 and to the right of 10 there is a square with 12.A net with a top row of three squares containing numbers 1, 1, and 3 from left to right. Below the square with 3 is a square with 2. To the right of the square with 2 is another square with a 3. To the right of this square with a 3 is another square with a 2.

    The three cubes are then arranged so that the numbers on their top faces are \(1\), \(2\) and \(3\), in some order. The sum of the three numbers on their bottom faces is

    1. \(8\)
    2. \(10\)
    3. \(12\)
    4. \(14\)
    5. \(16\)

  8. Suppose that \(x\) and \(y\) are integers satisfying \(-4\le x\le 11\) and \(9\le y\le 17\). Which of the following statements is correct for all such values of \(x\) and \(y\)?

    1. \(-2\le x-y\le 21\)
    2. \(6\le x-y\le 13\)
    3. \(-13\le x-y\le -6\)
    4. \(5\le x-y\le 28\)
    5. \(-21\le x-y\le 2\)

  9. There are \(28\) balls in a bag. Each ball is coloured \(1\) of \(7\) colours and has \(1\) of \(4\) patterns. No two balls have the same colour and pattern combination. Exactly \(3\) balls are removed from the bag, one at a time without replacement. What is the probability that one of the last two balls removed matches the colour of the first ball and the other matches the pattern of the first ball?

    1. \(\dfrac{1}{14}\)
    2. \(\dfrac{2}{39}\)
    3. \(\dfrac{28}{351}\)
    4. \(\dfrac{1}{13}\)
    5. \(\dfrac{4}{39}\)

  10. Arushi wrote an integer in each cell of a \(3\times 4\) grid so that the sum of the numbers in each row and in each column was the same. Some of the integers that Arushi wrote are shown in the grid while the others remain hidden.

    \(7\) \(3\)
    \(4\) \(c\)
    \(-1\) \(d\)

    The value of \(c-d\) is

    1. \(-1\)
    2. \(0\)
    3. \(5\)
    4. \(10\)
    5. \(13\)

Part C: Each correct answer is worth 8.

Each correct answer is an integer from 0 to 99, inclusive.

  1. Points \(P\) and \(Q\) are on the parabola with equation \(y = -3x^2 + 4x + 27\). The midpoint of \(PQ\) is \((0,0)\). If \(P\) lies above the \(x\)-axis, what is the \(y\)-coordinate of \(P\)?

  2. A rectangle with height \(20\) cm and width \(26\) cm is painted with \(n\) vertical strips and \(n\) horizontal strips. Each vertical strip has height \(20\) cm and width \(2\) cm. Each horizontal strip has height \(2\) cm and width \(26\) cm. Each vertical strip overlaps each horizontal strip. Vertical strips do not overlap one another, and horizontal strips do not overlap one another. The area of the painted portion of the rectangle is \(\frac{12}{13}\) of the area of the rectangle. What is the value of \(n\)?

  3. The string of digits \(123451234551234555\ldots\) is formed by alternately writing the digits \(1234\), in that order, and then writing some number of consecutive \(5\)s. There are exactly \(k\) consecutive \(5\)s immediately following the \(k\)th occurrence of \(1234\). If \(S\) is the sum of the first \(2026\) digits of the string, what is the sum of the digits of \(S\)?

  4. Adam collects \(n\) rocks, where \(100<n<300\). The ratio of the number of grey rocks collected to the number of spotted rocks collected is \(5:2\). The number of rocks that are both grey and spotted is \(m\), and is equal to the number of rocks that are neither grey nor spotted. What are the rightmost two digits of the number of possible ordered pairs of integers \((m,n)\)?

  5. The integers between \(1\) and \(10\) inclusive are to be arranged in a line so that if the integer \(a\) is divisible by the integer \(b\), then \(a\) appears to the left of \(b\). For example, \(6, 10, 8, 5, 9, 3, 4, 7, 2, 1\) is one such arrangement. Let \(N\) be the number of ways in which this can be done. What are the rightmost two digits of \(N\)?

Further Information

For students...

Thank you for writing the Fermat Contest!

Encourage your teacher to register you for Hypatia Contest which will be written in April.

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to