Problem 1.

For the continuous-time periodic signal

Assume that real numbers $a$ and $b$ satisfy
$$a b+\sqrt{a b+1}+\sqrt{a^{2}+b} \cdot \sqrt{b^{2}+a}=0$$
Find, with proof, the value of
$$a \sqrt{b^{2}+a}+b \sqrt{a^{2}+b}$$

Proof .

Problem 2.

Use the Fourier series analysis equation to calculate the coefficients $a_{k}$ for the continuous-time p

Let $d(k)$ denote the number of positive integer divisors of $k$. For example, $d(6)=4$ since 6 has 4 positive divisors, namely, $1,2,3$, and 6 . Prove that for all positive integers $n$,
$$d(1)+d(3)+d(5)+\cdots+d(2 n-1) \leq d(2)+d(4)+d(6)+\cdots+d(2 n)$$

Proof .

Directly calculate the covariation of x(t) with sin, cos

Problem 3.

Let $n \geq 2$ be an integer. Initially, the number 1 is written $n$ times on a board. Every minute, Vishal picks two numbers written on the board, say $a$ and $b$, erases them, and writes either $a+b$ or $\min \left{a^{2}, b^{2}\right}$. After $n-1$ minutes there is one number left on the board. Let the largest possible value for this final number be $f(n)$. Prove that
$$2^{n / 3}<f(n) \leq 3^{n / 3}$$

Proof .

Problem 4.

Let $x(t)$ be a periodic signal whose Fourier series coefficients are

a

Let $n$ be a positive integer. A set of $n$ distinct lines divides the plane into various (possibly unbounded) regions. The set of lines is called nice” if no three lines intersect at a single point. Acolouring” is an assignment of two colours to each region such that the first colour is from the set $\left{A_{1}, A_{2}\right}$, and the second colour is from the set $\left{B_{1}, B_{2}, B_{3}\right}$. Given a nice set of lines, we call it “colourable” if there exists a colouring such that

• no colour is assigned to two regions that share an edge;
• for each $i \in{1,2}$ and $j \in{1,2,3}$ there is at least one region that is assigned with both $\boldsymbol{A}{i}$ and $B{j}$
Determine all $n$ such that every nice configuration of $n$ lines is colourable.

Problem 5.

Let $A B C D E$ be a convex pentagon such that the five vertices lie on a circle and the five sides are tangent to another circle inside the pentagon. There are $\left(\begin{array}{l}5 \ 3\end{array}\right)=10$ triangles which can be formed by choosing 3 of the 5 vertices. For each of these 10 triangles, mark its incenter. Prove that these 10 incenters lie on two concentric circles.

The CMS wishes to thank the sponsors and partners who make our competitions program successful.

## CMO 2022

CMO 2022 takes place Thursday, March 10, 2022 starting at noon EST and running until 3pm EST.

Students get three hours to complete the CMO.  To maintain competition integrity, it runs concurrently no matter where the participant is located (therefore 1pm-4pm Atlantic, 9am-noon Pacific, etc.)

If you are an invited participant or involved as a teacher or proctor, please see our operational information page.

## CMO 2021

Official Results:Warren Bei CMO ChampionCongratulations to CMO Champion Warren Bei who is homeschooled in Vancouver, British Columbia! Warren has earned this year’s Canadian Mathematical Olympiad Cup and \$2,000 cash!

• View the official solutions.
• The top score this year was 29/35, the median score was 9/35.
• A total of 89 students from more than 48 different schools wrote the 2021 CMO.

2021 competitors can look up their own scores by giving their (6-digit) ID number here: .

## Matthew Brennan Award for Best CMO Solution

In 2021 the CMS and the mathematical community lost one of our most valuable members.  Matthew represented Canada twice at the International Mathematical Olympiad, earning a bronze medal in 2011 and a gold medal in 2012. He returned to the IMO as deputy leader observer in 2014 and 2017, and was the leader in 2019. Matt was passionate about Olympiad math, and had served on the Canadian Mathematical Olympiad committee since 2014. He also contributed extensively to problem creation and selection.

In Matthew’s honour, the CMS has created the “Matthew Brennan award for best CMO solution”. It is awarded every year to the student(s) who have written the best solution to a single problem on that year’s CMO. Their solution will be included in the official solutions, and they will receive a monetary prize.

This year, the winner of the Matthew Brennan Award for Best CMO solution goes to: Warren Bei

The CJMO is an olympiad style competition for younger students. Like for the CMO, the invitation to the CJMO is based on the results of the COMC and the Repêchage. Invitees will be the top students in grade 10 and below. Invitations to the CJMO will be sent out after the Repêchage has been graded.

## CJMO 2022

The 2022 CJMO exam will be written at the same time as the CMO – on Thursday, March 10th, 2022. Eligible students will be contacted via email.

## CJMO 2021

The 2021 CJMO exam was written at the same time as the CMO – on Thursday, March 11, 2021. There were 20 students competingOfficial Results:CJMO Champions

Congratulations to CJMO Champions Maggie Pang from Marc Garneau C.I. and William Zhao from Windemere Ranch Middle School.

## Past CJMO exams

2020: Examsolutions

2021: Examsolutions