2021 Euclid Contest

Wednesday, April 7, 2021
(in North America and South America)

Thursday, April 8, 2021
(outside of North America and South America)

©2021 University of Waterloo

Instructions

Time: $2\frac{1}{2}$ hours

Number of Questions: 10
Each question is worth 10 marks.

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.

Parts of each question can be of two types:

1. SHORT ANSWER parts indicated by
   • worth 3 marks each
   • full marks are given for a correct answer which is placed in the box
   • part marks are awarded if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate location in the answer booklet
   • marks awarded for completeness, clarity, and style of presentation
   • a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

• Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages.
• Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

1. 1. What is the value of $a$ for which $(a-1)+(2a-3)=14$?

   2. What are the two values of $c$ for which $(c^2-c)+(2c-3)=9$?

   3. Determine all values of $x$ for which ${\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{3}{2x^2}}=10$.

2. 1. What is the sum of the digits of the integer equal to $({10}^3+1{)}^2$?

   2. A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $1.50 and the price of each large cookie is $2.00. The price of each small cookie is increased by 10% and the price of each large cookie is increased by 5%. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie?

   3. Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages.

3. 1. In the diagram, $PQRS$ is a quadrilateral.

      

      What is its perimeter?

   2. In the diagram, $A$ has coordinates $(0,8)$.

      

      Also, the midpoint of $AB$ is $M(3,9)$ and the midpoint of $BC$ is $N(7,6)$. What is the slope of $AC$?

   3. The parabola with equation $y=-2x^2+4x+c$ has vertex $V(1,18)$. The parabola intersects the $y$-axis at $D$ and the $x$-axis at $E$ and $F$. Determine the area of $\mathrm{△}DEF$.

4. 1. If $3(8^x)+5(8^x)=2^{61}$, what is the value of the real number $x$?

   2. For some real numbers $m$ and $n$, the list $3n^2$, $m^2$, $2(n+1{)}^2$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$.

5. 1. Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$:

      Step 1: Reflect the point in the $x$-axis.

      Step 2: Translate the resulting point 2 units upwards.

      Step 3: Reflect the resulting point in the $y$-axis.

      As she does this, the point $(3,5)$ moves to $(3,-5)$, then to $(3,-3)$, and then to $(-3,-3)$.

      Chinara then starts with a different point $S_0$. She applies the three-step process $\mathcal{P}$ to the point $S_0$ and obtains the point $S_1$. She then applies $\mathcal{P}$ to $S_1$ to obtain the point $S_2$. She applies $\mathcal{P}$ four more times, each time using the previous output of $\mathcal{P}$ to be the new input, and eventually obtains the point $S_6(-7,-1)$. What are the coordinates of the point $S_0$?

   2. In the diagram, $ABDE$ is a rectangle, $\mathrm{△}BCD$ is equilateral, and $AD$ is parallel to $BC$.

      

      Also, $AE=2x$ for some real number $x$.

      1. Determine the length of $AB$ in terms of $x$.

      2. Determine positive integers $r$ and $s$ for which ${\displaystyle \frac{AC}{AD}}=\sqrt{{\displaystyle \frac{r}{s}}}$.

6. 1. Suppose that $n>5$ and that the numbers $t_1,t_2,t_3,\dots ,t_{n-2},t_{n-1},t_n$ form an arithmetic sequence with $n$ terms. If $t_3=5$, $t_{n-2}=95$, and the sum of all $n$ terms is 1000, what is the value of $n$?

      (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, $3,5,7,9$ are the first four terms of an arithmetic sequence.)

   2. Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6\sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is $30+30\sqrt{2}$. Determine all possible values for $a$.

      (A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, $3$, $-6$, $12$, $-24$ are the first four terms of a geometric sequence.)

7. 1. A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. What is the probability that, when he stops, there is at least 1 red ball and at least 1 green ball on the table?

   2. Suppose that $f(a)=2a^2-3a+1$ for all real numbers $a$ and $g(b)={\log }_{\frac{1}{2}}b$ for all $b>0$. Determine all $\theta$ with $0\le \theta \le 2\pi$ for which $f(g(\sin \theta ))=0$.

8. 1. Five distinct integers are to be chosen from the set $\{1,2,3,4,5,6,7,8\}$ and placed in some order in the top row of boxes in the diagram.

      

      Each box that is not in the top row then contains the product of the integers in the two boxes connected to it in the row directly above. Determine the number of ways in which the integers can be chosen and placed in the top row so that the integer in the bottom box is $9953280000$.

   2. Prove that the integer ${\displaystyle \frac{(1!)(2!)(3!)\cdots (398!)(399!)(400!)}{200!}}$ is a perfect square. (In this fraction, the numerator is the product of the factorials of the integers from 1 to 400, inclusive.)

9. 

   1. Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K,L)$ for which $K^2+3L^2=a^2+b^2-ab$.

   2. Prove that, for all integers $K$ and $L$, there is at least one pair of integers $(a,b)$ for which $K^2+3L^2=a^2+b^2-ab$.

   3. Prove that, for all integers $a$ and $b$, there is at least one pair of integers $(K,L)$ for which $K^2+3L^2=a^2+b^2-ab$.

10. 

    1. In the diagram, eleven circles of four different sizes are drawn.

       

       Each circle labelled $W$ has radius 1, each circle labelled $X$ has radius 2, the circle labelled $Y$ has radius 4, and the circle labelled $Z$ has radius $r$. Each of the circles labelled $W$ or $X$ is tangent to three other circles. The circle labelled $Y$ is tangent to all ten of the other circles. The circle labelled $Z$ is tangent to three other circles. Determine positive integers $s$ and $t$ for which $r={\displaystyle \frac{s}{t}}$.

    2. Suppose that $c$ is a positive integer. Define $f(c)$ to be the number of pairs $(a,b)$ of positive integers with $c<a<b$ for which two circles of radius $a$, two circles of radius $b$, and one circle of radius $c$ can be drawn so that

       • each circle of radius $a$ is tangent to both circles of radius $b$ and to the circle of radius $c$, and

       • each circle of radius $b$ is tangent to both circles of radius $a$ and to the circle of radius $c$,

       as shown.

       

       Determine all positive integers $c$ for which $f(c)$ is even.

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