2021 Fermat Contest
(Grade 11)

Thursday, February 23, 2021
(in North America and South America)

Wednesday, February 24, 2021
(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. A rectangle has width $2\text{ cm}$ and length $3\text{ cm}$.

   

   The area of the rectangle is

   1. $2{\text{ cm}}^2$
   2. $9{\text{ cm}}^2$
   3. $5{\text{ cm}}^2$
   4. $36{\text{ cm}}^2$
   5. $6{\text{ cm}}^2$

2. The expression $2+3\times 5+2$ equals

   1. $19$
   2. $27$
   3. $35$
   4. $17$
   5. $32$

3. The number equal to 25% of 60 is

   1. $10$
   2. $15$
   3. $20$
   4. $12$
   5. $18$

4. When $x=2021$, the value of ${\displaystyle \frac{4x}{x+2x}}$ is

   1. ${\displaystyle \frac{3}{4}}$
   2. ${\displaystyle \frac{4}{3}}$
   3. $2021$
   4. $2$
   5. $6$

5. Which of the following integers cannot be written as a product of two integers, each greater than 1?

   1. $6$
   2. $27$
   3. $53$
   4. $39$
   5. $77$

6. A square piece of paper has a dot in its top right corner and is lying on a table. The square is folded along its diagonal, then rotated ${90}^{\circ }$ clockwise about its centre, and then finally unfolded, as shown.

   

   The resulting figure is

   1. 
   2. 
   3. 
   4. 
   5. 

7. For which of the following values of $x$ is $x$ greater than $x^2$ ?

   1. $x=-2$
   2. $x=-\frac{1}{2}$
   3. $x=0$
   4. $x=\frac{1}{2}$
   5. $x=2$

8. The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer?

   1. $5$
   2. $7$
   3. $6$
   4. $8$
   5. $9$

9. The line with equation $y=2x-6$ is translated upwards by 4 units. (That is, every point on the line is translated upwards by 4 units, forming a new line.) The $x$-intercept of the resulting line is

   1. $3$
   2. $\frac{3}{2}$
   3. $4$
   4. $1$
   5. $2$

10. If $3^x=5$, the value of $3^{x+2}$ is

    1. $10$
    2. $25$
    3. $2187$
    4. $14$
    5. $45$

Part B: Each correct answer is worth 6.

11. In the sum shown, $P$, $Q$ and $R$ represent three different single digits.

    

    The value of $P+Q+R$ is

    1. $13$
    2. $12$
    3. $14$
    4. $3$
    5. $4$

12. How many of the 20 perfect squares $1^2,2^2,3^2,\dots ,{19}^2,{20}^2$ are divisible by 9?

    1. $2$
    2. $3$
    3. $4$
    4. $5$
    5. $6$

13. In the diagram, each of $\mathrm{△}WXZ$ and $\mathrm{△}XYZ$ is an isosceles right-angled triangle.

    

    The length of $WX$ is $6\sqrt{2}$. The perimeter of quadrilateral $WXYZ$ is closest to

    1. $18$
    2. $20$
    3. $23$
    4. $25$
    5. $29$

14. Natascha cycles 3 times as fast as she runs. She spends 4 hours cycling and 1 hour running. The ratio of the distance that she cycles to the distance that she runs is

    1. $12:1$
    2. $7:1$
    3. $4:3$
    4. $16:9$
    5. $1:1$

15. Let $a$ and $b$ be positive integers for which $45a+b=2021$. The minimum possible value of $a+b$ is

    1. $44$
    2. $82$
    3. $85$
    4. $86$
    5. $130$

16. If $n$ is a positive integer, the notation $n!$ (read “$n$ factorial”) is used to represent the product of the integers from 1 to $n$. That is, $n!=n(n-1)(n-2)\cdots (3)(2)(1)$. For example, $4!=4(3)(2)(1)=24$ and $1!=1$. If $a$ and $b$ are positive integers with $b>a$, the ones (units) digit of $b!-a!$ cannot be

    1. $1$
    2. $3$
    3. $5$
    4. $7$
    5. $9$

17. The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$?

    1. $15$
    2. $16$
    3. $17$
    4. $18$
    5. $19$

18. In the diagram, $\mathrm{△}PQR$ is an isosceles triangle with$PQ=PR$. Semi-circles with diameters $PQ$, $QR$ and $PR$ are drawn.

    

    The sum of the areas of these three semi-circles is equal to 5 times the area of the semi-circle with diameter $QR$. The value of $\cos (\mathrm{\angle }PQR)$ is

    1. $\frac{1}{3}$
    2. $\frac{1}{\sqrt{8}}$
    3. $\frac{1}{\sqrt{12}}$
    4. $\frac{1}{\sqrt{15}}$
    5. $\frac{1}{\sqrt{10}}$

19. The real numbers $x$, $y$ and $z$ satisfy the three equations $$\begin{array}{rl}x+y& =7\\ xz& =-180\\ (x+y+z{)}^2& =4\end{array}$$ If $S$ is the sum of the two possible values of $y$, then $-S$ equals

    1. $56$
    2. $14$
    3. $36$
    4. $34$
    5. $42$

20. In the diagram, $PRTY$ and $WRSU$ are squares.

    

    Point $Q$ is on $PR$ and point $X$ is on $TY$ so that $PQXY$ is a rectangle. Also, point $T$ is on $SU$, point $W$ is on $QX$, and point $V$ is the point of intersection of $UW$ and $TY$, as shown. If the area of rectangle $PQXY$ is 30, the length of $ST$ is closest to

    1. $5$
    2. $5.25$
    3. $5.5$
    4. $5.75$
    5. $6$

Part C: Each correct answer is worth 8.

21. A function, $f$, has $f(2)=5$ and $f(3)=7$. In addition, $f$ has the property that $$f(m)+f(n)=f(mn)$$ for all positive integers $m$ and $n$. (For example, $f(9)=f(3)+f(3)=14$.) The value of $f(12)$ is

    1. $17$
    2. $35$
    3. $28$
    4. $12$
    5. $25$

22. An unpainted cone has radius 3 cm and slant height 5 cm.

    

    The cone is placed in a container of paint. With the cone’s circular base resting flat on the bottom of the container, the depth of the paint in the container is 2 cm. When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as ${\displaystyle \frac{p}{q}}$ where $p$ and $q$ are positive integers with no common divisor larger than 1. What is the value of $p+q$?

    (The lateral surface of a cone is its external surface not including the circular base. A cone with radius $r$, height $h$, and slant height $s$ has lateral surface area equal to $\pi rs$.) 

    1. $59$
    2. $61$
    3. $63$
    4. $65$
    5. $67$

23. In Figure 1, three unshaded dots are arranged to form an equilateral triangle, as shown.

    

    Figure 2 is formed by arranging three copies of Figure 1 to form the outline of a larger equilateral triangle and then filling the resulting empty space with 1 shaded dot.

    

    For each integer $n>2$, Figure $n$ is formed by first arranging three copies of Figure $n-1$ to form the outline of a larger equilateral triangle and then filling the resulting empty space in the centre with an inverted triangle of shaded dots.

    

    Hide/Reveal Description of Figure 3 for Question 21

    An equilateral triangle with a base of eight unshaded dots. The second row is seven dots: unshaded, shaded, unshaded, shaded, unshaded, shaded, unshaded. The third row is six dots: unshaded, unshaded, shaded, shaded, unshaded, unshaded. The fourth row is five dots: unshaded, shaded, shaded, shaded, and unshaded. The fifth row is four unshaded dots.

    The sixth row is three dots: unshaded, shaded, unshaded. The seventh row is two unshaded dots. The top row is one unshaded dot.

    ---

    The smallest value of $n$ for which Figure $n$ includes at least 100 000 shaded dots is

    1. $8$
    2. $9$
    3. $10$
    4. $11$
    5. $12$

24. A pair of real numbers $(a,b)$ with $a^2+b^2\le \frac{1}{4}$ is chosen at random. If $p$ is the probability that the curves with equations $y=a{x}^2+2bx-a$ and $y=x^2$ intersect, then $100p$ is closest to

    1. $65$
    2. $69$
    3. $53$
    4. $57$
    5. $61$

25. Let $N$ be the number of triples $(x,y,z)$ of positive integers such that $x<y<z$ and $xyz=2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot {11}^2\cdot {13}^2\cdot {17}^2\cdot {19}^2$. When $N$ is divided by 100, the remainder is

    1. $28$
    2. $88$
    3. $8$
    4. $68$
    5. $48$

Further Information

For students...

Thank you for writing the Fermat Contest!

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

Visit our website cemc.uwaterloo.ca to find

• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
• Information about careers in and applications of mathematics and computer science

For teachers...

Visit our website cemc.uwaterloo.ca to

• Obtain information about future contests
• Look at our free online courseware for high school students
• Learn about our face-to-face workshops and our web resources
• Subscribe to our free Problem of the Week
• Investigate our online Master of Mathematics for Teachers
• Find your school's contest results