2025 Fermat Contest
(Grade 11)

Wednesday, February 26, 2025
(in North America and South America)

Thursday, February 27, 2025
(outside of North American and South America)

©2024 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.

Part A: Each correct answer is worth 5.

1. The value of \(2-0+2\times 5\) is

   1. \(-8\)
   2. \(10\)
   3. \(0\)
   4. \(12\)
   5. \(2\)

2. In a survey, each student chose their favourite day of the week. In the graph, the percentages of those surveyed who selected each day of the week is shown.

   

   If \(3000\) students were surveyed, how many people chose Friday as their favourite day of the week?

   1. \(400\)
   2. \(600\)
   3. \(700\)
   4. \(500\)
   5. \(800\)

3. How many integer values of \(x\) satisfy the inequality \(2<x<14\)?

   1. \(10\)
   2. \(7\)
   3. \(13\)
   4. \(14\)
   5. \(11\)

4. Rachel, Christophe and Alfonzo are paid to organize some books. In total, they are paid \(\$50\). Alfonzo is paid \(\$14\). Rachel is paid twice what Christophe is paid. How much is Christophe paid?

   1. \(\$10\)
   2. \(\$12\)
   3. \(\$14\)
   4. \(\$16\)
   5. \(\$18\)

5. In the diagram, points \(P\), \(Q\), \(R\), and \(S\) are at the intersection of gridlines in a \(5\times 5\) grid of \(1 \times 1\) squares.

   

   The area of rectangle \(PQRS\) is

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

6. The current calendar year, \(2025\), is a perfect square. In \(n\) years from 2025, the calendar year will again be a perfect square. The smallest possible value of \(n\) is

   1. \(2025\)
   2. \(100\)
   3. \(9\)
   4. \(46\)
   5. \(91\)

7. In the diagram, the target shown has three scoring areas. An arrow that hits the centre circle is worth \(10\) points, an arrow that hits the shaded middle ring is worth \(5\) points, and an arrow that hits the outer ring is worth \(1\) point.

   

   Three arrows are shot and each hits the target. Which of the following cannot be the total score for the three arrows?

   1. \(16\)
   2. \(11\)
   3. \(13\)
   4. \(7\)
   5. \(20\)

8. The average of \(15\) integers is \(18\). The average of \(5\) of these integers is \(12\). What is the average of the other \(10\) integers?

   1. \(15\)
   2. \(21\)
   3. \(24\)
   4. \(18\)
   5. \(12\)

9. If \(x^2-y^2=72\) and \(x-y=12\), the value of \(x+y\) is

   1. \(8\)
   2. \(60\)
   3. \(9\)
   4. \(72\)
   5. \(6\)

10. There are \(186\) students on a class trip. Each student is placed into exactly one of \(50\) groups. Each group has exactly \(3\) students or exactly \(4\) students. There are \(m\) groups that have \(3\) students and \(n\) groups that have \(4\) students. The value of \(m-n\) is

    1. \(-22\)
    2. \(-10\)
    3. \(-14\)
    4. \(-26\)
    5. \(-18\)

Part B: Each correct answer is worth 6.

11. Twelve lightbulbs are in a row. All lightbulbs are initially turned off. Angie flips the switch for every 2nd lightbulb. Then Bilal flips the switch for every 3rd lightbulb. Finally, Chenxhui flips the switch for every 4th lightbulb. At the end of this process, how many lightbulbs are turned on?

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

12. In the graph, the five points show the money earned versus time worked for each of five employees. Each was paid a different but fixed number of dollars per hour. Which letter represents the employee who was paid the most money per hour?

    

    Hide/Reveal Description of Diagram for Question 12

    A graph with horizontal axis Time (hours) and vertical axis Money Earned ($). There are \(14\) equally spaced vertical lines, including the vertical axis, and \(15\) equally spaced horizontal lines, including the horizontal axis. Each of the letters \(A\) through \(E\) marks a point where two lines intersect. Counting horizontal lines from the bottom up and vertical lines from left to right, not including the axes, the letters are located at the following intersections:

    • Letter \(A\): 2nd vertical line and 10th horizontal line
    • Letter \(B\): 4th vertical line and 6th horizontal line
    • Letter \(C\): 6th vertical line and 12th horizontal line
    • Letter \(D\): 8th vertical line and 4th horizontal line
    • Letter \(E\): 10th vertical line and 8th horizontal line

    ---

    1. \(A\)
    2. \(B\)
    3. \(C\)
    4. \(D\)
    5. \(E\)

13. If the equation \((x+2)(x+t)= x^2 + bx + 12\) is true for all real numbers \(x\), the value of \(b\) is

    1. \(6\)
    2. \(8\)
    3. \(10\)
    4. \(12\)
    5. \(14\)

14. A substance doubles its volume every minute. At \(\mbox{9:00 a.m.}\) a small amount of the substance was placed in a large empty container. At \(\mbox{9:20 a.m.}\) the same day, the container became full. At what time was the container one-quarter full?

    1. \(\mbox{9:15 a.m.}\)
    2. \(\mbox{9:10 a.m.}\)
    3. \(\mbox{9:16 a.m.}\)
    4. \(\mbox{9:04 a.m.}\)
    5. \(\mbox{9:18 a.m.}\)

15. The hundreds digit of the smallest five-digit positive integer that is divisible by \(12\), \(13\), \(14\) and \(15\) is

    1. \(2\)
    2. \(7\)
    3. \(1\)
    4. \(9\)
    5. \(3\)

16. In the diagram, \(ABCD\) is a rectangle with area \(224\). Semi-circles with diameters \(AD\) and \(BC\) are drawn inside the rectangle.

    

    If the shortest distance between the semi-circles is \(2\), the area of the shaded region is closest to

    1. \(50\)
    2. \(55\)
    3. \(60\)
    4. \(65\)
    5. \(70\)

17. A tennis tournament starts with \(8\) players. Francesca is equally likely to play against any of the other \(7\) players in her first match. If Francesca plays against Dominique or Estella, the probability that Francesca wins is \(\frac{2}{5}\). If Francesca plays against any of the other \(5\) players, the probability that she wins is \(\frac{3}{4}\). What is the probability that Francesca wins her first match?

    1. \(\dfrac{23}{50}\)
    2. \(\dfrac{29}{50}\)
    3. \(\dfrac{3}{5}\)
    4. \(\dfrac{13}{20}\)
    5. \(\dfrac{2}{3}\)

18. In a \(2000 \mbox{ m}\) race, Arturo, Morgan and Henri run at constant but different speeds. Arturo finishes \(200 \mbox{ m}\) ahead of Morgan and \(290 \mbox{ m}\) ahead of Henri. If Morgan and Henri each continue at their same speeds, how far ahead of Henri will Morgan finish?

    1. \(90\mbox{ m}\)
    2. \(100\mbox{ m}\)
    3. \(110\mbox{ m}\)
    4. \(120\mbox{ m}\)
    5. \(130\mbox{ m}\)

19. The lines with equations \(y=mx+7\), \(y=2\), \(x=0\), and \(y=0\) form a trapezoid with area 3. If \(m>0\), what is the value of \(m\)?

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

20. Some integers \(m\) with \(1<m<100\,000\) have the property that the product of the digits of \(m\) is equal to \(200\). If \(N\) is the number of such integers \(m\), what is the integer formed by the rightmost two digits of \(N\)?

    1. \(17\)
    2. \(27\)
    3. \(37\)
    4. \(47\)
    5. \(57\)

Part C: Each correct answer is worth 8.

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

21. The area of a right-angled triangle is \(54\mbox{ cm}^2\). The side lengths of the triangle are \(a\) cm, \(b\) cm, and \(c\) cm, where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\). What is the value of \(c\)?

22. The triple \((x,y,z)\) of integers satisfies the following system of equations: \[\begin{align*} 2^x + 2^y + 3^{z-1} & = 2259\\ 2^{x+y} + 3^z & = 7073\\ 2^x+2^y+3^z & = 6633\end{align*}\] If \(P\) is equal to the product \(xyz\), what is the integer formed by the rightmost two digits of \(P\)?

23. When two ants work together they can build an anthill in 24 minutes. When the bigger ant works alone, an anthill can be built in 14 minutes less than when the smaller ant works alone. How many minutes does it take the smaller ant to build an anthill when working alone?

24. Suppose that \(f(x)=x^4+px^3+qx^2+rx+s\) for some real numbers \(p\), \(q\), \(r\), \(s\). In addition, \(f(1)=59\), \(f(2)=118\) and \(f(3)=177\). If \(T=f(9)+f(-5)\), what is the sum of the digits of the integer equal to \(T\)?

25. A sequence \(a_1\), \(a_2\),\(\ldots\) has \(a_1 = 1\), \(a_2 = 3\) and \(a_n = -a_{n-1}+a_{n-2}\) for each integer \(n \geq 3\). For example, \(a_3=-a_2+a_1=-2\). How many of the 2025 integers \((a_1)^2\), \((a_2)^2\), \((a_3)^2\), \(\ldots\), \((a_{2025})^2\) are divisible by 2025?

Further Information

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