2024 Euclid Contest

Wednesday, April 3, 2024
(in North America and South America)

Thursday, April 4, 2024
(outside of North American and South America)

©2024 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 2 or 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.

Questions

1. 1. If $x=2$, what is the value of ${\displaystyle \frac{x^4+3x^2}{x^2}}$ ?

   2. In the diagram, $\mathrm{△}ABC$ is right-angled at $B$. Also, $AB=10$, $BC=t-1$, and $AC=t+1$. What is the value of $t$?

      

   3. Suppose that ${\displaystyle \frac{2}{y}}+{\displaystyle \frac{3}{2y}}=14$. Determine the value of $y$.

2. 1. In a sequence with six terms, each term after the second is the sum of the previous two terms. If the fourth term is $13$ and the sixth term is $36$, what is the first term?

   2. For some real number $r\ne 0$, the sequence $5r$, $5r^2$, $5r^3$ has the property that the second term plus the third term equals the square of the first term. What is the value of $r$?

   3. Jimmy wrote four tests last week. The average of his marks on the first, second and third tests was $65$. The average of his marks on the second, third and fourth tests was $80$. His mark on the fourth test was $2$ times his mark on the first test. Determine his mark on the fourth test.

3. 1. The graph of the equation $y=r(x-3)(x-r)$ intersects the $y$-axis at $(0,48)$. What are the two possible values of $r$?

   2. A bicycle costs $\mathrm{\$}B$ before taxes. If the sales tax were $13\mathrm{\%}$, Annemiek would pay a total that is $\mathrm{\$}24$ higher than if the sales tax were $5\mathrm{\%}$. What is the value of $B$?

   3. The function $f$ has the following three properties:

      • $f(1)=3$.

      • $f(2n)=(f(n){)}^2$ for all positive integers $n$.

      • $f(2m+1)=3f(2m)$ for all positive integers $m$.

      Determine the value of $f(2)+f(3)+f(4)$.

4. 1. In the diagram, $AB$ is perpendicular to $CD$ (with $B$ on $CD$), $CP$ is perpendicular to $AD$ (with $P$ on $AD$), and $N$ is the point of intersection of $AB$ and $CP$.

      

      Also, $\mathrm{\angle }ADB=45\mathrm{°}$, $AB=12$, and $CB=6$. What is the area of $\mathrm{△}APN$?

   2. In the diagram, the line with equation $y=-3x+6$ crosses the $x$-axis at $A$ and the $y$-axis at $B$. Suppose that $m>0$ and that the line with equation $y=mx+1$ crosses the $y$-axis at $D$ and intersects the line with equation $y=-3x+6$ at the point $C$.

      

      If $O$ is the origin and the area of $\mathrm{△}ACD$ is $\frac{1}{2}$ of the area of $\mathrm{△}ABO$, determine the coordinates of $C$.

5. 1. In the diagram, rectangle $ABCD$ is divided into four smaller rectangles by the lines $PQ$ and $RS$, which intersect at $X$.

      

      The areas of these smaller rectangles are, in some order, $2$, $6$, $3$, and $a$. What are the three possible values of $a$?

   2. Suppose that the parabola with equation $y=x^2-4tx+5t^2-6t$ has two distinct $x$-intercepts. Determine the value of $t$ for which the distance between these $x$-intercepts is as large as possible.

6. 1. There are $M$ integers between $10000$ and $100000$ that are multiples of $21$ and whose units (ones) digit is $1$. What is the value of $M$?

   2. There are $N$ students who attend Strickland S.S., where $500<N<600$. Among these $N$ students, $\frac{2}{5}$ are in the physics club and $\frac{1}{4}$ are in the math club. In the physics club, there are $2$ times as many students who are not in the math club as there are students who are in the math club. Determine the number of students who are not in either club.

7. 1. Arun and Bella run around a circular track, starting from diametrically opposite points. Arun runs clockwise around the track and Bella runs counterclockwise. Arun and Bella run at constant, but different, speeds. They meet for the first time after Arun has run $100$ m. They meet for the second time after Bella runs $150$ m past their first meeting point. What is the length of the track?

   2. Determine all angles $\theta$ with $0\mathrm{°}\le \theta \le 360\mathrm{°}$ for which $4^{1+{\cos }^3\theta }=2^{2-\cos \theta }\cdot 8^{{\cos }^2\theta }$.

8. 1. In the diagram, the circle with centre $A$ has radius $4$, and $A$ lies on the circle which has centre $B$ and radius $3$. The line passing through $A$ and $B$ lies along the diameter of each circle and is perpendicular to $MN$ at $T$. Also, $MN$ is tangent to the larger circle, and $MC$ and $NC$ are each tangent to both circles at points $D$, $E$, $F$, and $G$, as shown. Determine the area of $\mathrm{△}MNC$.

      

   2. Determine all triples $(x,y,z)$ of real numbers that are solutions to the following system of equations: $$\begin{array}{rl}{\log }_{9}x+{\log }_{9}y+{\log }_{3}z& =2\\ {\log }_{16}x+{\log }_{4}y+{\log }_{16}z& =1\\ {\log }_{5}x+{\log }_{25}y+{\log }_{25}z& =0\end{array}$$

9. An ant walks along the $x$-axis by taking a sequence of steps of length $1$. Some, all or none of these steps are in the positive $x$-direction; some, all or none of these steps are in the negative $x$-direction. The ant begins at $x=0$, takes a total of $n$ steps, and ends at $x=d$. For each such sequence, let $c$ be the number of times that the ant changes direction.

   1. Determine the number of different sequences of steps for which $n=9$ and $d=5$.

   2. Suppose that $n=9$ and $d=3$. Determine the number of sequences for which $c$ is even.

   3. Determine the number of pairs $(d,n)$ of integers with $1\le n\le 2024$ and $d\ge 0$ for which $c$ is even for exactly half of the sequences of $n$ steps that end at $x=d$.

10. Suppose that $s$ and $t$ are real numbers with $0<s\le 1$ and $0<t\le 1$. Points $A(-1,0)$, $B(0,4)$ and $C(1,0)$ form $\mathrm{△}ABC$. Points $S(s,0)$ and $T(-t,0)$ lie on $AC$. Point $P$ lies on $AB$ and point $Q$ lies on $BC$, with neither $P$ nor $Q$ at a vertex of $\mathrm{△}ABC$. Line segments $SP$ and $TQ$ intersect at $X$ and partition $\mathrm{△}ABC$ into four regions. For some such pairs $(s,t)$ of real numbers and points $P$ and $Q$, the line segments $SP$ and $TQ$ in fact partition $\mathrm{△}ABC$ into four regions of equal area. We call such a pair $(s,t)$ a balancing pair.

    1. Suppose that $(s,t)$ is a balancing pair with $s=1$ and that line segments $SP$ and $TQ$ partition $\mathrm{△}ABC$ into four regions of equal area. Determine the coordinates of $P$.

    2. Prove that there exist real numbers $d$, $e$, $f$, and $g$ for which all balancing pairs $(s,t)$ satisfy an equation of the form $$s^2+t^2=dst+es+ft+g$$ and determine the values of $d$, $e$, $f$, and $g$.

    3. Determine an infinite family of distinct pairs of rational numbers $(s,t)$ with $0<s\le t\le 1$ that satisfy the equation in (b).

Further Information

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