2020 Fermat Contest
(Grade 11)

Tuesday, February 25, 2020
(in North America and South America)

Wednesday, February 26, 2020
(outside of North American and South America)

©2020 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. 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.
7. 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.
8. Diagrams are not drawn to scale. They are intended as aids only.
9. When your supervisor tells you to begin, you will have sixty minutes of working time.
10. 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 points \(O(0, 0)\), \(P(0,3)\), \(Q\), and \(R(5,0)\) form a rectangle, as shown.

   

   The coordinates of \(Q\) are

   1. \((5,5)\)
   2. \((5,3)\)
   3. \((3,3)\)
   4. \((2.5,1.5)\)
   5. \((0,5)\)

2. The value of \(3\times 2020 + 2\times 2020 - 4\times 2020\) is

   1. \(6060\)
   2. \(4040\)
   3. \(8080\)
   4. \(0\)
   5. \(2020\)

3. For every real number \(x\), the expression \((x+1)^2 - x^2\) is equal to

   1. \(2x+1\)
   2. \(2x-1\)
   3. \((2x+1)^2\)
   4. \(-1\)
   5. \(x+1\)

4. Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is \(3, 14, 25, 36, \ldots\). A number that will appear in Ewan’s sequence is

   1. \(113\)
   2. \(111\)
   3. \(112\)
   4. \(110\)
   5. \(114\)

5. The value of \(\sqrt{\dfrac{\phantom{\big(}\!\!\sqrt{81}+\sqrt{81}}{2}}\) is

   1. \(3\)
   2. \(18\)
   3. \(27\)
   4. \(81\)
   5. \(162\)

6. Anna thinks of an integer.

   • It is not a multiple of three.

   • It is not a perfect square.

   • The sum of its digits is a prime number.

   The integer that Anna is thinking of could be

   1. \(12\)
   2. \(14\)
   3. \(16\)
   4. \(21\)
   5. \(26\)

7. In the diagram, \(WXY\) is a straight angle.

   

   What is the average (mean) of \(p\), \(q\), \(r\), \(s\), and \(t\)?

   1. \(30\)
   2. \(36\)
   3. \(60\)
   4. \(72\)
   5. \(45\)

8. If \(2^n = 8^{20}\), what is the value of \(n\)?

   1. \(10\)
   2. \(60\)
   3. \(40\)
   4. \(16\)
   5. \(17\)

9. The figure consists of five squares and two right-angled triangles.

   

   Hide/Reveal Description of Diagram for Question 9

   Together, these squares and triangles form a crooked tower, where each shape that is joined to another shape shares a common side as follows: a large square with area 32 is the base of the tower. Stacked on top of the square is a right-angled triangle whose right angle lines up with the left side of the square. A second large square (unshaded and unmarked) is joined to the hypotenuse of the triangle, and thus leans to the right.

   A third smaller square with area 8 is joined to the left side of the triangle. Now a second right-angled triangle is stacked on top of the leaning large square so that its right angle lines up with the right side of the leaning square. A fourth smaller square with area 5 is joined to the right side of the triangle. Finally, a large shaded square is stacked on the hypotenuse of the right-angled triangle.

   ---

   The areas of three of the squares are 5, 8 and 32, as shown. What is the area of the shaded square?

   1. \(35\)
   2. \(45\)
   3. \(29\)
   4. \(19\)
   5. \(75\)

10. Positive integers \(s\) and \(t\) have the property that \(s(s - t) = 29\). What is the value of \(s+t\)? 

    1. \(1\)
    2. \(28\)
    3. \(57\)
    4. \(30\)
    5. \(29\)

Part B: Each correct answer is worth 6.

11. In the \(5 \times 5\) grid shown, 15 cells contain X’s and 10 cells are empty.

    X	X	X	X
    X	X	X		X
    X	X
    X	X		X
    		X	X

    Any X may be moved to any empty cell. What is the smallest number of X’s that must be moved so that each row and each column contains exactly three X’s?

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

12. Harriet ran a 1000 m course in 380 seconds. She ran the first 720 m of the course at a constant speed of 3 m/s. She ran the remaining part of the course at a constant speed of \(v\) m/s. What is the value of \(v\)?

    1. \(2\)
    2. \(1.5\)
    3. \(3\)
    4. \(1\)
    5. \(4.5\)

13. In the list \(2,x,y,5\), the sum of any two adjacent numbers is constant. The value of \(x-y\) is

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

14. In Rad’s garden there are exactly 30 red roses, exactly 19 yellow roses, and no other roses. How many of the yellow roses does Rad need to remove so that \(\frac{2}{7}\) of the roses in the garden are yellow?

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

15. Suppose that \(N=3x+4y+5z\), where \(x\) equals 1 or \(-1\), and \(y\) equals 1 or \(-1\), and \(z\) equals 1 or \(-1\). How many of the following statements are true?

    • \(N\) can equal 0.
    • \(N\) is always odd.
    • \(N\) cannot equal 4.
    • \(N\) is always even.
    1. \(0\)
    2. \(1\)
    3. \(2\)
    4. \(3\)
    5. \(4\)

16. Suppose that \(x\) and \(y\) are real numbers with \(-4 \leq x \leq -2\) and \(2 \leq y \leq 4\). The greatest possible value of \(\dfrac{x + y}{x}\) is

    1. \(1\)
    2. \(-1\)
    3. \(-\frac{1}{2}\)
    4. \(0\)
    5. \(\frac{1}{2}\)

17. In the diagram, \(\triangle PQR\) is right-angled at \(Q\) and point \(S\) is on \(PR\) so that \(QS\) is perpendicular to \(PR\).

    

    If the area of \(\triangle PQR\) is 30 and \(PQ=5\), the length of \(QS\) is

    1. \(\frac{60}{13}\)
    2. \(5\)
    3. \(\frac{30}{13}\)
    4. \(4\)
    5. \(3\)

18. Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point for a tie. If \(S\) is the sum of the points of the four teams after the tournament is complete, which of the following values can \(S\) not equal?

    1. \(13\)
    2. \(17\)
    3. \(11\)
    4. \(16\)
    5. \(15\)

19. When \((3+2x+x^2)(1+mx+m^2x^2)\) is expanded and fully simplified, the coefficient of \(x^2\) is equal to 1. What is the sum of all possible values of \(m\)?

    1. \(-\frac{4}{3}\)
    2. \(-\frac{2}{3}\)
    3. \(0\)
    4. \(\frac{2}{3}\)
    5. \(\frac{4}{3}\)

20. A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What is the probability that the top face has an odd number of dots on it?

    1. \(\frac{4}{7}\)
    2. \(\frac{1}{2}\)
    3. \(\frac{13}{27}\)
    4. \(\frac{11}{21}\)
    5. \(\frac{3}{7}\)

Part C: Each correct answer is worth 8.

21. In the diagram, the central circle contains the number 36.

    

    Positive integers are to be written in the eight empty circles, one number in each circle, so that the product of the three integers along any straight line is 2592. If the nine integers in the circles must be all different, what is the largest possible sum of these nine integers?

    1. \(160\)
    2. \(176\)
    3. \(178\)
    4. \(195\)
    5. \(216\)

22. Suppose that \(x\) and \(y\) are real numbers that satisfy the two equations: \[\begin{aligned} x^2+3xy+y^2 & =909 \\ 3x^2+xy+3y^2 & = 1287\end{aligned}\] What is a possible value for \(x+y\)?

    1. \(27\)
    2. \(39\)
    3. \(29\)
    4. \(92\)
    5. \(41\)

23. There are real numbers \(a\) and \(b\) for which the function \(f\) has the properties that \(f(x) = ax+b\) for all real numbers \(x\), and \(f(bx+a)=x\) for all real numbers \(x\). What is the value of \(a+b\)?

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

24. In the diagram, the circle with centre \(X\) is tangent to the largest circle and passes through the centre of the largest circle. The circles with centres \(Y\) and \(Z\) are each tangent to the other three circles, as shown.

    

    The circle with centre \(X\) has radius 1. The circles with centres \(Y\) and \(Z\) each have radius \(r\). The value of \(r\) is closest to

    1. \(0.93\)
    2. \(0.91\)
    3. \(0.95\)
    4. \(0.87\)
    5. \(0.89\)

25. Three real numbers \(x\), \(y\), \(z\) are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of \(x-y\) and \(x-z\) is greater than \(-\frac{1}{2}\) and less than \(\frac{1}{2}\) ?

    1. \(\frac{3}{4}\)
    2. \(\frac{7}{12}\)
    3. \(\frac{1}{4}\)
    4. \(\frac{1}{2}\)
    5. \(\frac{2}{3}\)

Further Information

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