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2024 Fermat Contest
(Grade 11)

Wednesday, February 28, 2024
(in North America and South America)

Thursday, February 29, 2024
(outside of North American and South America)

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©2023 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 value of \(3\left(\frac 53-\frac 13\right)\) is

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

  2. If \(x=2\), the value of \(4x^2-3x^2\) is

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

  3. How many solid \(1\times 1\times 1\) cubes are required to make a solid \(2\times 2\times 2\) cube?

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

  4. Shuxin begins with \(10\) red candies, \(7\) yellow candies, and \(3\) blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate?

    1. \(11\)
    2. \(7\)
    3. \(17\)
    4. \(20\)
    5. \(14\)

  5. Square \(PQRS\) is divided into \(16\) smaller congruent squares, as shown.

    Of the 16 smaller squares, 2 squares have their full area shaded and 8 squares have half their area shaded.

    What fraction of \(PQRS\) is shaded?

    1. \(\frac 13\)
    2. \(\frac 16\)
    3. \(\frac 12\)
    4. \(\frac 14\)
    5. \(\frac 38\)

  6. How many integers are greater than \(\sqrt{15}\) and less than \(\sqrt{50}\)?

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

  7. The line with equation \(y=3x+6\) is reflected in the \(y\)-axis. What is the \(x\)-intercept of the new line?

    1. \(2\)
    2. \(-2\)
    3. \(6\)
    4. \(-6\)
    5. \(\frac 12\)

  8. If \(10^n = 1000^{20}\), what is the value of \(n\)?

    1. \(1000\)
    2. \(60\)
    3. \(2000\)
    4. \(300\)
    5. \(102\)

  9. In the diagram, a semi-circle has centre \(O\) and diameter \(AC\). Also, \(B\) is a point on the circumference such that \(\angle BAC = 25\degree\).

    The measure of \(\angle BOC\) is

    1. \(60\degree\)
    2. \(55\degree\)
    3. \(45\degree\)
    4. \(50\degree\)
    5. \(65\degree\)

  10. In a photograph, Aristotle, David, Flora, Munirah, and Pedro are seated in a random order in a row of \(5\) chairs. If David is seated in the middle of the row, what is the probability that Pedro is seated beside him?

    1. \(\frac 15\)
    2. \(\frac 14\)
    3. \(\frac 12\)
    4. \(\frac 25\)
    5. \(\frac 35\)

Part B: Each correct answer is worth 6.

  1. Figure 1 shows an arrangement of \(3\) lines with \(1\) intersection point, and Figure 2 shows an arrangement of \(3\) lines with \(3\) intersection points.

    Figure 1 has three different lines all passing through a common point. Figure 2 has three lines placed so that each pair of lines intersects at a single point and these three points of intersection form a triangle.

    What is the maximum number of intersection points that can appear in an arrangement of \(4\) lines?

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

  2. The numbers \(5\), \(6\), \(10\), \(17\), and \(21\) are rearranged so that the sum of the first three numbers is equal to the sum of the last three numbers. Which number is in the middle of this rearrangement?

    1. \(5\)
    2. \(6\)
    3. \(10\)
    4. \(17\)
    5. \(21\)

  3. For some integers \(m\) and \(n\), the expression \((x+m)(x+n)\) is equal to a quadratic expression in \(x\) with a constant term of \(-12\). Which of the following cannot be a value of \(m\)?

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

  4. In the diagram, point \(C\) is on side \(BD\) of quadrilateral \(ABDE\). Also, \(AB\) and \(ED\) are perpendicular to \(BD\), \(\angle ACB = 60\degree\), \(\angle CAE = 45\degree\), and \(\angle AEC = 45\degree\).

    If \(AB = \sqrt{3}\), what is the perimeter of quadrilateral \(ABDE\,\)?

    1. \(1+\sqrt{2}+2\sqrt{3}\)
    2. \(2+2\sqrt{2}+\sqrt{3}\)
    3. \(1+2\sqrt{2}+2\sqrt{3}\)
    4. \(2+\sqrt{2}+2\sqrt{3}\)
    5. \(2+2\sqrt{2}+2\sqrt{3}\)

  5. Anila’s grandmother wakes up at the same time every day and follows this same routine:

    If Anila’s grandmother woke up \(5\) minutes ago, what will she be doing in \(197\) hours?

    1. Getting her coffee
    2. Having a shower
    3. Doing some yoga
    4. Going for a walk
    5. Calling her granddaughter

  6. Francesca put one of the integers \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\) in each of the nine squares of the \(3\) by \(3\) grid shown. No integer is used twice. She calculated the product of the three integers in each row and wrote the products to the right of the corresponding rows. She calculated the product of the integers in each column and wrote the products below the corresponding columns. Finally, she erased the integers from the nine squares.

    From top to bottom, the numbers to the right of the rows are 56, 135, and 48. From left to right, the numbers below the columns are 21, 108, and 160. The letter N is placed in the middle square of the bottom row.

    Which integer was in the square marked \(N\,\)?

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

  7. Each of the variables \(a\), \(b\), \(c\), \(d\), and \(e\) represents a positive integer with the properties that \[\begin{align*} b+d &> a + d\\ c+e &> b+e\\ b+d&= c\\ a+c &= b+e\end{align*}\] Which of the variables has the greatest value?

    1. \(a\)
    2. \(b\)
    3. \(c\)
    4. \(d\)
    5. \(e\)

  8. Suppose that \(x\) and \(y\) are real numbers that satisfy the two equations \[\begin{align*} 3x+2y &= 6\\ 9x^2+4y^2&= 468\end{align*}\] What is the value of \(xy\)?

    1. \(-36\)
    2. \(-12\)
    3. \(-32\)
    4. \(-120\)
    5. \(-32\)

  9. Three standard six-sided dice are rolled. The sum of the three numbers rolled, \(S\), is determined. The probability that \(S>5\) is closest to

    1. \(0.97\)
    2. \(0.95\)
    3. \(0.94\)
    4. \(0.96\)
    5. \(0.98\)

  10. A cylinder contains some water. A solid cone with the same height and half the radius of the cylinder is submerged into the water until the circular face of the cone lies flat on the circular base of the cylinder, as shown.

    Once this is done, the depth of the water is half of the height of the cylinder. If the cone is then removed, the depth of the water will be what fraction of the height of the cylinder?

    (The volume of a cylinder with radius \(r\) and height \(h\) is \(\pi r^2h\) and the volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2h\).)

    1. \(\frac{3}{16}\)
    2. \(\frac{41}{96}\)
    3. \(\frac{5}{16}\)
    4. \(\frac{3}{8}\)
    5. \(\frac{7}{16}\)

Part C: Each correct answer is worth 8.

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

  1. A \(3 \times 3\) table starts with every entry equal to \(0\) and is modified using the following steps:
    (i) adding \(1\) to all three numbers in any row;
    (ii) adding \(2\) to all three numbers in any column.

    After step (i) has been used a total of \(a\) times and step (ii) has been used a total of \(b\) times, the table appears as shown.

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

    What is the value of \(a+b\)?

  2. Four distinct integers \(a\), \(b\), \(c\), and \(d\) are chosen from the set \(\{1,2,3,4,5,6,7,8,9,10\}\). What is the greatest possible value of \(ac+bd-ad-bc\)?

  3. In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AB\) and point \(E\) is on \(BC\) so that \(DE\) is perpendicular to \(BC\), \(BE=AC\), \(BD=120\), and \(DE+BC=288\). What is the length of \(DE\,\)?

  4. The integer \(N\) is the smallest positive integer that is a multiple of \(2024\), has more than \(100\) positive divisors (including \(1\) and \(N\)), and has fewer than \(110\) positive divisors (including \(1\) and \(N\)). What is the sum of the digits of \(N\)?

  5. A sequence of \(11\) positive real numbers, \(a_1\), \(a_2\), \(a_3\), \(\ldots\), \(a_{11}\), satisfies \(a_1=4\) and \(a_{11}=1024\) and \(a_n+a_{n-1}=\tfrac{5}{2}\sqrt{a_{n}\cdot a_{n-1}}\) for every integer \(n\) with \(2 \leq n \leq 11\). For example when \(n=7\), \(a_7+a_{6}=\tfrac{5}{2}\sqrt{a_{7} \cdot a_{6}}\). There are \(S\) such sequences. What are the rightmost two digits of \(S\)?

Further Information

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