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2026 Canadian Team Mathematics Contest

Individual Problems (45 minutes)

Important Notes

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) previously stored information such as formulas, programs, notes, etc., (iv) a computer algebra system, (v) dynamic geometry software.
Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi+1$ and $1-\sqrt{2}$ are simplified exact numbers.

Problems

What is the value of $3^2 + 2^3$?
Including $2026$, how many different four-digit positive integers can be formed from the digits of $2026$? Note each such positive integer should have one $6$, two $2$s, and one $0$.
On the planet Cemtece, each Cemtece-day has $16$ Cemtece-hours, and each Cemtece-hour has $15$ Earth-minutes. How many Cemtece-days does one Earth-day have?
If $a^4=3$, what is the value of $\left(a^2+\dfrac{1}{a^2}\right)^{\!2}$?
How many of the positive divisors of $900$ are perfect squares?

An integer is a perfect square if it is equal to the square of an integer. For example, $16$ is a perfect square, and $7$ is not a perfect square.
Real numbers $x$, $y$, and $z$ satisfy $x=11z-2y$, $z=2x-3y$, and $x yz \neq 0$. What is the value of $\dfrac{x}{y}$?
In regular hexagon $ABCDEF$, trapezoids $AGJF$, $CHGB$, and $EJHD$ are congruent, with $GH=HJ=GJ=3$. What is the area of the hexagon?
The parabola with equation $y=-2x^2+16x+k$ intersects the line with equation $y=kx+14$ at points $A$ and $B$. If the midpoint of $AB$ is $\left(\frac{7}{2},21\right)$ what is the distance between $A$ and $B$?
How many words of length $18$ are there, consisting only of $A$s and $B$s such that
- there are never $4$ (or more) consecutive $A$s, and
- there are never $2$ (or more) consecutive $B$s, and
- every two $B$s have at least two $A$s between them?
For example, one such word is $BAABAAABAABAABAABA$.
The sequence $a_1, a_2, a_3, \dots$ is an arithmetic sequence with common difference $45$ and $a_1=8$. The sequence $b_1, b_2, b_3, \dots$ is an arithmetic sequence with common difference $d$ and $b_1=233$, where $d$ is an integer. Among integers that appear in both of the sequences and are greater than $2026$, the second smallest integer is $2258$. What is the sum of all possible values of $d$?

Team Problems (45 minutes)

Important Notes

Calculating devices are not permitted.
Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi+1$ and $1-\sqrt{2}$ are simplified exact numbers.

Problems

How many bricks with dimensions $1\times 1\times 2$ are stacked to form the solid rectangular prism shown?

The front face of the prism is a square and is made up of three 1 by 1 squares and three 1 by 2 rectangles arranged into three rows as follows: a 1 by 1 square followed by a 1 by 2 rectangle form the top row, and also the bottom row, and a 1 by 2 rectangle followed by a 1 by 1 square form the middle row.

The right face is a rectangle and is made up of four 1 by 1 squares and four 1 by 2 rectangles arranged into three rows as follows: a 1 by 1 square followed by a 1 by 2 rectangle followed by a 1 by 1 square form the top row, and also the bottom row, and two 1 by 2 rectangles form the middle row.
If $4 \le a \le 20$, what is the maximum possible value of $\dfrac{100}{a}$?
A square and a circle have the same area. If the radius of the circle is $2$, what is the side length of the square?
Consider angles $x\degree$, $2x\degree$, $3x\degree$, and $y\degree$ in the diagram below.

Given that $x\degree + y\degree = 60\degree$, what is the value of $y$?
$N$ is a three-digit positive integer with a middle digit of zero. The sum of the other two digits is $11$. If the digits are reversed, the integer formed is greater than the original integer, $N$, by $495$. What is the value of $N$?
Zendaya is drawing on graph paper. Starting at $(0,0)$, with a red pen, she repeatedly traces along grid lines $1$ unit to the right and then $1$ unit up. Starting at $(0,0)$, with a blue pen, she repeatedly traces along grid lines $2$ units up and then $2$ units right. Not counting the origin, the blue and red lines touch each other $2026$ times. What is the area of the region enclosed by the red lines and blue lines?
The fraction $\dfrac{65}{49}$ can be written in the form $1+\dfrac{1}{a+\dfrac{1}{b}}$, where $a$ and $b$ are positive integers. What are $a$ and $b$?
Four circular wheels, $A$, $B$, $C$, and $D$ turn around their centres. They have radii $1$ cm, $5$ cm, $2$ cm and $3$ cm respectively. When wheel $A$ is turned, it turns $B$, which turns $C$, which turns $D$. There is no slipping throughout this process.

If wheel $D$ turns at $1$ revolution per second, at how many revolutions per second does $A$ turn?
The prime numbers $a, b$, and $c$ have the property that $2a+5b+10c=155$ and $c-b=4$. What are the values of $a$, $b$, and $c$?
Using the keyboard below, sequences of letters can be formed starting at Q and ending at M where each letter in the sequence other than Q is either immediately below or immediately to the right (on the keyboard) of the letter before it in the sequence. For example, the only two letters that can follow D in a sequence are F and X. How many such sequences of letters can be formed?
A command called swap switches two letters in a word. For example, using the input TABLE, the command swap(1,4) switches the 1^st and 4^th letters in the word, to obtain the output LABTE. Using the input CANDY, the following program is run. What is the output?
```
Input word
Repeat 2026 times:
    swap(1,2)
    swap(3,5)
    swap(2,4)
Output final result
```
Note that each of the three swap commands is executed $2026$ times.
For a positive integer $n$, let $S_n$ be the sum of the digits of $n$. There is exactly one positive integer $c$ that satisfies $c=11 \times S_c$. What is $c$?
In square $ABCD$, which has side length $12$, $FG$ is parallel to $BC$, $EF$ is half of $AB$, and $BD$ divides $\triangle GEF$ into two regions with the same area. What is the exact value of $\dfrac{AB}{AE}$?
The integer $d>1$ has the property that $332$, $456$, and $549$ have the same remainder when divided by $d$. What is the value of $d$?
A chemist has $3$ beakers, each containing a (well-mixed) mixture of acid and water:
- Bottle $A$ contains $40$ mL, $10 \%$ of which is acid.
- Bottle $B$ contains $50$ mL, $20 \%$ of which is acid.
- Bottle $C$ contains $50$ mL, $30 \%$ of which is acid.
She uses some of the mixture from each of the bottles to create a mixture with volume $60$ mL, $25 \%$ of which is acid. She then mixes the remaining contents of the three bottles to create a new mixture. What percentage of the new mixture is acid?
The geometric sequence with $n$ terms $t_1$, $t_2$, $\dots$, $t_n$ satisfies $t_1 t_n = 3$. The product of all $n$ terms of the sequence is equal to $6561$. What is the value of $n$?

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 with common ratio $2$.
The integer $x$ is randomly generated from the range $0\le x < 360$, with each of the $360$ integers equally likely to be chosen. What is the probability that $4 \sin x \cos x + 2 \sin x > 2\sqrt{3} \cos x + \sqrt{3}$?

We take $x$ to be in degrees and not radians for this question.
Among integers $N$ of the form $N = 42xyz + 21yz + 14xz + 6xy + 2x + 3y + 7z + 1$, where $x$, $y$, $z$ are distinct positive integers, what is the smallest value of $N$ that is a product of distinct primes?

$30$ is a product of distinct primes, since $30 = 2 \times 3 \times 5$, while $20$ is not a product of distinct primes, since $20=2 \times 2 \times 5$.
In $\triangle ABC$, $AB=AC$, and $M$ is the midpoint of $BC$. Point $D$ is chosen on $AM$ such that $MD=1$, $AD=10$, and $\angle BDC = 3 \angle BAC$. What is the perimeter of $\triangle ABC$?
Given that \[(\log _ { 2} 3) (\log_{3^2} 2)+(\log _ { 2^2} 3)(\log_{3^3} 2)+(\log _ { 2^3} 3) (\log_{3^4}2)+\cdots+ (\log_{2^n} 3)(\log_{3^{n+1}} 2)=\dfrac{7}{8},\] what is the value of $n$? (The sum has $n$ terms.)
One ant is located at $(1,2)$ and another ant is located as $(6,5)$. Each minute, both ants move simultaneously $1$ unit in the plane. The ant initially at $(1,2)$ moves $1$ unit to the right or $1$ unit up, each with probability $\frac{1}{2}$. The ant initially at $(6,5)$ moves $1$ unit to the left or $1$ unit down, each with probability $\frac{1}{2}$. What is the probability that, after each ant has moved $4$ times, they are at the same point?
Paula writes $n$ consecutive positive integers that have a sum of $475$, where $n>1$. How many possible values are there for $n$?
For which range of real numbers $k>0$ do the equations $x^2 + 2x +y^2 = (k-1)(k+1)$ and
$x^2 - 2x + y^2 = \left(\dfrac{1}{k} - 1\right)\left(\dfrac{1}{k} + 1\right)$ have a common real solution $(x,y)$?
In a $4 \times 4$ grid consisting of $16$ unit squares, $4$ squares are coloured red, while the other $12$ are coloured white. Two colourings are said to be equivalent if one can be obtained from the other by doing a sequence of $90\degree$ rotations. How many inequivalent colourings are there?
What is the maximum value of the expression \[1-\frac{32}{\left(1+\sqrt{1+\sqrt{1+x}}\right)^3} - \frac{32}{\left(1-\sqrt{1+\sqrt{1+x}}\right)^3} -\frac{16}{\left(1+\sqrt{1+\sqrt{1+x}}\right)^4} - \frac{16}{\left(1-\sqrt{1+\sqrt{1+x}}\right)^4}\] as $x$ ranges over all real numbers for which the expression is defined?

Relays

Relay #1

Seat a

Given that $x+3=5$, what is the value of $2x+4$?

Seat b

Let $t$ be TNYWR.
A triangle has vertices $A(-5,3),~B(3,3),$ and $C(2,t)$. What is the area of $\triangle ABC$?

Seat c

Let $t$ be TNYWR.
The quadratic function $f(x) = (x-b)(x-3)$ satisfies $f(1)=t$, where $b$ is a fixed real number. What is the value of $f(5)$?

Relay #2

Seat a

$S$ is a point on the number line shown such that $PS=\left(\dfrac{3}{2}\right) PR$ and $QR=RS$. What number is located at point $Q$?

Four points marked on a number line. From left to right, the points are labelled P, Q, R, and S. The number 4 is located at point P and the number 32 is located at point R.

Seat b

Let $t$ be TNYWR.
Line $l_1$ has equation $6x-3y+t=0$ and line $l_2$ has equation $ax-2y+24=0$. Lines $l_1$ and $l_2$ have the same $x$-intercept. What is the slope of $l_2$?

Seat c

Let $t$ be TNYWR.
What is the largest solution of $2(x-3)\sqrt{x}=tx$?

Relay #3

Seat a

Given that $AB$ is a straight line, what is the value of $x$?

A straight angle is made up of three angles measuring 23 degrees, 2x degrees, and 61 degrees.

Seat b

Let $t$ be TNYWR.
Gurpreet initially has $\$d$. Gurpreet lends $\frac{1}{2}$ of his money to his sister, and donated $\frac{1}{5}$ of the original amount to charity. If he then has $\$t$ remaining, what is the value of $d$?

Seat c

Let $t$ be TNYWR.
If the perimeter of the rectangle shown is $t$, what is the value of $x$?

The top side of the rectangle has length equal to 5y minus x; the bottom side has length equal to 2y plus 5x; and the right side has length 17.