2026 Euclid Contest

Tuesday, March 31, 2026
(in North America and South America)

Wednesday, April 1, 2026
(outside of North American and South America)

University of Waterloo Logo

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:

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
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.

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 our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

NOTE:

Please read the instructions for the contest.
Write all answers in the answer booklet provided.
For questions marked , place your answer in the appropriate box in the answer booklet and show your work.
For questions marked , provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution.
Diagrams are not drawn to scale. They are intended as aids only.
While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions, and specific marks may be allocated for these steps. For example, while your calculator might be able to find the $x$-intercepts of the graph of an equation like $y=x^{3}-x$, you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

Questions

1. What is the integer $t$ for which $\dfrac{2t}{3} + \dfrac{3t}{2} = 26$?
2. What is the integer $x$ for which $\dfrac{3+x}{4} = \dfrac{6+x}{8}$ ?
3. Suppose that $y > 0$ and $\sqrt{3^2+4^2+12^2} = \sqrt{3^2+4^2} + \sqrt{y^2}$. Determine the value of $y$.
1. The sum of the digits of the positive integer $2026$ is $2+0+2+6=10$. What is the smallest integer $n> 2026$ whose digits also have a sum of 10?
2. The product of the digits of the integer $313$ is $3 \cdot 1 \cdot 3 = 9$. How many integers between $100$ and $999$, including $313$, have the property that the product of their digits is $9$?
3. The sum of $x$, $3x$ and $4y$ is equal to $48$. The average of $x$ and $y$ is equal to $3x$. Determine the ordered pair $(x,y)$.
1. What is the smallest positive integer $n$ for which $\dfrac{9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3}{n}$ is equal to $k^3$ for some integer $k$?
2. What is the ordered pair $(a,b)$ that satisfies both of the equations $3^{a+b} = 27$ and $a-b = -5$?
3. For some real number $c$, the parabola with equation $y = -x^2 + 7x + c$ intersects the $x$-axis at points $P(10, 0)$ and $Q$. If the parabola intersects the $y$-axis at $R$, determine the area of $\triangle PQR$.
1. In January, Rebecca measured the temperature in Yellowknife every day at 11:00 a.m. The average of these $31$ temperatures was $-20^\circ\mbox{C}$. The average of the temperatures from the first $21$ days was $-15^\circ\mbox{C}$. What was the average of the temperatures from the last $10$ days?
2. McKayla runs to her grandmother's house and then runs home along the same straight road. The route from McKayla's house, $M$, to her grandmother's house, $G$, is on flat ground from $M$ to $H$, and then uphill from $H$ to $G$, as shown in the cross-section below. The distance from $M$ to $H$ to $G$ is $10$ km. (That is, $MH+HG=10$ km.)
  
  McKayla runs on flat ground at $12$ km/h, uphill at $10$ km/h, and downhill at $15$ km/h. It takes $54$ minutes for her to run from $M$ to $H$ to $G$. Determine the number of minutes that it takes for her to run from $G$ to $H$ to $M$.
1. Two fair dice, called $D_1$ and $D_2$, each have six faces. $D_1$ has the numbers $1$, $2$, $3$, $4$, $5$, $6$ on its faces. $D_2$ has a $1$ on some of its faces and a $2$ on its remaining faces. When $D_1$ and $D_2$ are rolled, the probability that the sum of the numbers on the top faces is a prime number is $\dfrac{23}{36}$. How many faces on $D_2$ have the number $1$ on them?
2. In the diagram, square $ABCD$ has $A$ and $B$ on the $x$-axis and $C$ and $D$ below the $x$-axis on the parabola with equation $y = x^2 - 4$.
  
  Determine the area of $ABCD$, writing your answer in the form $r - \sqrt{t}$ for some positive integers $r$ and $t$.
1. Xander, Yasmin and Zhe each have a rope. Xander's rope is $10$ m long. Yasmin's rope is $n\%$ longer than Xander's rope. Zhe's rope is $(2n)\%$ longer than Yasmin's rope. Zhe's rope is $(3.14n)\%$ longer than Xander's rope. If $n > 0$, what is the value of $n$?
2. In the diagram, quadrilateral $ABCD$ has $AB=AD=4$. Also, $\angle ABC = 45\degree$ and $\angle CDA = 135\degree$.
  
  Determine the exact value of $BC - CD$.
1. In a garden, there are roses and carnations, and no other kind of flower. Each flower is either yellow or white. Half of the yellow flowers are roses, $\dfrac{1}{4}$ of the roses are yellow, and $\dfrac{1}{4}$ of all of the flowers are white carnations. What fraction of all of the flowers are yellow roses?
2. The function $f$ is defined for every real number $x > 0$ in the following way: \[f(x) = \begin{cases} 0 & \text{if $0 < x \leq 1$}\\ 1 + f(\log_2 x) & \text{if $x > 1$} \end{cases}\] Determine the number of positive integers $n$ for which $f(n) = 4$.
1. In the diagram, the circle has centre $O$ and radius $2$. Points $A$, $B$, $U$, and $V$ are on the circle so that $AB$ and $UV$ are diameters. Point $C$ is outside the circle so that $AC$ is tangent to the circle at $A$ and so that $BC$ intersects the circle at $U$. If $BU = 2UC$, determine the area of $\triangle CUV$.
2. Determine all positive integers $k$ for which there are exactly $100$ non-congruent triangles that have positive integer side lengths, one side length equal to $k$, perimeter equal to $3k$, and one angle with measure greater than $90\degree$.
1. Suppose that $\triangle XYZ$ has $XY = XZ$ and $YZ = b$. If the area of $\triangle XYZ$ is $40$ and the perimeter of $\triangle XYZ$ is $32$, determine a cubic polynomial $f$ with integer coefficients and with the property that $f(b) = 0$.
2. Consider positive real numbers $A$ and $P$. If there exists an isosceles triangle with area $A$ and perimeter $P$, prove that there exist at most two non-congruent isosceles triangles with area $A$ and perimeter $P$.
3. Determine positive integers $A$ and $P$ for which there exist two non-congruent isosceles triangles $XYZ$ with $XY=XZ$, integer side lengths less than $300$, area $A$, and perimeter $P$, and with the property that the length of $YZ$ in exactly one of the two triangles is a multiple of $7$.
Suppose that $n$ is a positive integer with $n \geq 5$. An arrangement $a_1, a_2, \ldots, a_{n-1}, a_n$ of the $n$ positive integers $1, 2, \ldots, {n\!-\!1}, n$ is said to have an internal peak at position $t$ (with $2 \leq t \leq {n-1}$) if $a_{t-1} < a_t$ and $a_t > a_{t+1}$. For example, the arrangement $2, 3, 7, 1, 5, 4, 6, 8$ of the integers $1, 2, 3, 4, 5, 6, 7, 8$ has an internal peak at position $3$, an internal peak at position $5$, and no other internal peaks.
1. Determine the number of arrangements of $1, 2, 3, 4, 5$ that have an internal peak at position $3$ and no other internal peaks.
2. Suppose that $n \geq 5$. Determine, in terms of $n$, an expression in closed form for the number of arrangements of $1, 2, \ldots, {n\!-\!1}, n$ that have an internal peak at position $2$ and no other internal peaks.
3. Suppose that $n \geq 5$. Determine, in terms of $n$, an expression in closed form for the number of arrangements of $1, 2, \ldots, {n\!-\!1}, n$ that have an internal peak at position $3$ and no other internal peaks.
Note 1: Depending on your approach, the following formula might be useful.

For every positive integer $r$, \[\sum_{k=0}^r k2^k = 0 \cdot 2^0 + 1 \cdot 2^1 + 2 \cdot 2^2 + \cdots + r 2^r = (r-1)2^{r+1} + 2\]

Note 2: If you are unsure what is meant by "closed form" in parts (b) and (c), consider the three equal expressions in Note 1 as an example:
- The expression $\displaystyle \sum_{k=0}^rk2^k$ is not a closed form.
- The expression $0 \cdot 2^0 + 1 \cdot 2^1 + 2 \cdot 2^2 + \cdots + r 2^r$ is not a closed form.
- The expression $(r-1)2^{r+1}+2$ is a closed form.

Further Information

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