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 the largest possible value for this final number be \( f(n) \). Prove that
\[
2^{n/3} < f(n) \leq 3^{n/3}.
\]
|
[] |
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 \( \{A_1, A_2\} \), and the second colour is from the set \( \{B_1, B_2, B_3\} \). Given a nice set of lines, we call it “colourable” if there exists a colouring such that
(a) no colour is assigned to two regions that share an edge;
(b) for each \( i \in \{1, 2\} \) and \( j \in \{1, 2, 3\} \) there is at least one region that is assigned with both \( A_i \) and \( B_j \).
Determine all \( n \) such that every nice configuration of \( n \) lines is colourable.
|
[] |
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 incenters lie on two concentric circles.
|
[] |
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 that Victor needs to guarantee this?
|
[] |
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 class buys tickets to the concert?
(Assume that friendship is mutual; if student \( A \) is close friends with student \( B \), then \( B \) is close friends with \( A \).)
|
[] |
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 \( CF \) again at \( X \), \( Y \), and \( Z \) respectively. Prove the following inequality:
\[
\frac{AH}{DX} + \frac{BH}{EY} + \frac{CH}{FZ} \geq 3.
\]
|
[] |
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
\[
f(\text{divs}(m)) = \text{divs}(n).
\]
Prove that for any such \( f \), there are finitely many \( f \)-cool integers.
(The notation \( f[S] \) for some set \( S \) denotes the set \( \{ f(s) : s \in S \} \).)
|
[] |
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 \( n \) and \( k \)) such that there is guaranteed to be a set of \( m \) cities, no two of which are directly connected by a road?
|
[] |
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 irreducible in \( \mathbb{Q} \) if it cannot be factored into two non-constant polynomials with rational coefficients.)
|
[] |
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 up to scan a specific subgrid \([a, b] \times [c, d]\) with \( 1 \leq a \leq b \leq M \) and \( 1 \leq c \leq d \leq N \). Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up \( Q \) detectors, which may only be run simultaneously after all \( Q \) detectors are ready. In terms of \( M \) and \( N \), what is the minimum \( Q \) required to guarantee your crew can determine the location of Blackboard’s legendary treasure?
|
[] |
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.
\item If two points \( P_1 \) and \( P_2 \) are marked, draw the line connecting \( P_1 \) and \( P_2 \).
\item If two non-parallel lines \( \ell_1 \) and \( \ell_2 \) are drawn, mark the intersection of \( \ell_1 \) and \( \ell_2 \).
\item If a line \( \ell \) is drawn, draw a line parallel to \( \ell \) that is at distance 1 away from \( \ell \) (note that two such lines may be drawn).
\end{itemize}
Prove that it is possible to mark the orthocenter of \( ABC \) using these operations.
|
[] |
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 procedure: At the time of the update, if \( a \) is voting for \( b \), and \( b \) is voting for \( c \), then \( a \) updates their vote to \( c \). (Note that \( a \), \( b \), and \( c \) need not be distinct; if \( b = c \), then \( a \)'s vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds).
They repeat this updating procedure for \( n \) rounds. Prove that at this time, all \( n \) players will unanimously vote for the same person.
|
[] |
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 coefficients. Prove that there exist reflexive polynomials \( q(x), r(x) \) with integer coefficients such that
\[
(1 + x + x^2 + \cdots + x^{\ell - 1}) p(x) = q(x) + x^\ell r(x).
\]
|
[] |
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 of the incircles of \( BAE \) and \( CAD \) intersect at \( Z \). Prove that \( Z \) lies on the common chord of \( \Gamma \) and the circumcircle of \( AXY \).
|
[] |
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 \( r \in S \) such that the ant is currently located at one of the corners of \( r \), say \( c \), and move the ant to one of the two corners of \( r \) adjacent to \( c \).
Suppose that after a finite number of operations, the ant ends up at the top-right corner of \( R \). Prove that some rectangle \( r \in S \) was chosen in at least two operations.
|
[] |
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