id string | source string | problem string | solutions list |
|---|---|---|---|
CMO-2022-3 | https://cms.math.ca/wp-content/uploads/2022/03/2022CMO-exam-en.pdf | 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 \( \min(a^2, b^2) \). After \( n - 1 \) minutes there is one number left on the board. Let th... | [] |
CMO-2022-4 | https://cms.math.ca/wp-content/uploads/2022/03/2022CMO-exam-en.pdf | 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. A “colouring” is an assignment of two colours to each region such that the first colour is from the set \( \{A_... | [] |
CMO-2022-5 | https://cms.math.ca/wp-content/uploads/2022/03/2022CMO-exam-en.pdf | Let \( ABCDE \) 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 \( \binom{5}{3} = 10 \) 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 i... | [] |
CMO-2023-1 | https://cms.math.ca/wp-content/uploads/2023/03/2023CMO-exam-en.pdf | William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer \( m \) and ask William: “does \( m \) divide your number?”, to which William must answer truthfully. Victor continues asking these questions until he determines William’s number. What is the minimum number of questions ... | [] |
CMO-2023-2 | https://cms.math.ca/wp-content/uploads/2023/03/2023CMO-exam-en.pdf | There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too.
Is it possible that the entire... | [] |
CMO-2023-3 | https://cms.math.ca/wp-content/uploads/2023/03/2023CMO-exam-en.pdf | An acute triangle is a triangle that has all angles less than \( 90^\circ \) (\( 90^\circ \) is a Right Angle). Let \( ABC \) be an acute triangle with altitudes \( AD \), \( BE \), and \( CF \) meeting at \( H \). The circle passing through points \( D \), \( E \), and \( F \) meets \( AD \), \( BE \), and \( CF \) ag... | [] |
CMO-2023-4 | https://cms.math.ca/wp-content/uploads/2023/03/2023CMO-exam-en.pdf | Let \( f(x) \) be a non-constant polynomial with integer coefficients such that \( f(1)
e 1 \). For a positive integer \( n \), define \( \text{divs}(n) \) to be the set of positive divisors of \( n \).
A positive integer \( m \) is \( f \)-cool if there exists a positive integer \( n \) for which
\[
f(\text{divs}(m)... | [] |
CMO-2023-5 | https://cms.math.ca/wp-content/uploads/2023/03/2023CMO-exam-en.pdf | A country with \( n \) cities has some two-way roads connecting certain pairs of cities. Someone notices that if the country is split into two parts in any way, then there would be at most \( kn \) roads between the two parts (where \( k \) is a fixed positive integer). What is the largest integer \( m \) (in terms of ... | [] |
CMO-2024-1 | https://cms.math.ca/wp-content/uploads/2024/03/CMO2024-problems.pdf | Let \( ABC \) be a triangle with incenter \( I \). Suppose the reflection of \( AB \) across \( CI \) and the reflection of \( AC \) across \( BI \) intersect at a point \( X \). Prove that \( XI \) is perpendicular to \( BC \).
(The incenter is the point where the three angle bisectors meet.) | [] |
CMO-2024-2 | https://cms.math.ca/wp-content/uploads/2024/03/CMO2024-problems.pdf | Jane writes down 2024 natural numbers around the perimeter of a circle. She wants the 2024 products of adjacent pairs of numbers to be exactly the set \( \{1!, 2!, \ldots, 2024!\} \). Can she accomplish this? | [] |
CMO-2024-3 | https://cms.math.ca/wp-content/uploads/2024/03/CMO2024-problems.pdf | Let \( N \) be the number of positive integers with 10 digits \( d_9d_8 \cdots d_1d_0 \) in base 10 (where \( 0 \leq d_i \leq 9 \) for all \( i \) and \( d_9 > 0 \)) such that the polynomial
\[
d_9x^9 + d_8x^8 + \cdots + d_1x + d_0
\]
is irreducible in \( \mathbb{Q} \). Prove that \( N \) is even.
(A polynomial is irr... | [] |
CMO-2024-4 | https://cms.math.ca/wp-content/uploads/2024/03/CMO2024-problems.pdf | Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of an \( M \times N \) ( \( 2 \leq M, N \) ) grid-structured island. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure. For each detector, you can set it ... | [] |
CMO-2024-5 | https://cms.math.ca/wp-content/uploads/2024/03/CMO2024-problems.pdf | Initially, three non-collinear points, \( A, B, \) and \( C \), are marked on the plane. You have a pencil and a double-edged ruler of width 1. Using them, you may perform the following operations:
\begin{itemize}
\item Mark an arbitrary point in the plane.
\item Mark an arbitrary point on an already drawn line.
... | [] |
CMO-2025-1 | https://cms.math.ca/wp-content/uploads/2025/03/CMO2025-problems.pdf | The \( n \) players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.
The players will update their votes via a series of rounds. In one round, each player \( a \) updates their vote, one at a time, according to the following pr... | [] |
CMO-2025-2 | https://cms.math.ca/wp-content/uploads/2025/03/CMO2025-problems.pdf | Determine all positive integers \( a, b, c, p \) where \( p \) and \( p + 2 \) are odd primes and
\[
2^a p^b = (p + 2)^c - 1.
\] | [] |
CMO-2025-3 | https://cms.math.ca/wp-content/uploads/2025/03/CMO2025-problems.pdf | A polynomial \( c_d x^d + c_{d-1} x^{d-1} + \cdots + c_1 x + c_0 \) with degree \( d \) is reflexive if there is an integer \( n \ge d \) such that \( c_i = c_{n-i} \) for every \( 0 \le i \le n \), where \( c_i = 0 \) for \( i > d \). Let \( \ell \ge 2 \) be an integer and \( p(x) \) be a polynomial with integer coeff... | [] |
CMO-2025-4 | https://cms.math.ca/wp-content/uploads/2025/03/CMO2025-problems.pdf | Let \( ABC \) be a triangle with circumcircle \( \Gamma \) and \( AB
e AC \). Let \( D \) and \( E \) lie on the arc \( BC \) of \( \Gamma \) not containing \( A \) such that \( \angle BAE = \angle DAC \). Let the incenters of \( BAE \) and \( CAD \) be \( X \) and \( Y \) respectively, and let the external tangents o... | [] |
CMO-2025-5 | https://cms.math.ca/wp-content/uploads/2025/03/CMO2025-problems.pdf | A rectangle \( R \) is divided into a set \( S \) of finitely many smaller rectangles with sides parallel to the sides of \( R \) such that no three rectangles in \( S \) share a common corner. An ant is initially located at the bottom-left corner of \( R \). In one operation, we can choose a rectangle \( r \in S \) su... | [] |
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