Olympiad problems

2025 Caucasus Mathematical Olympiad, 1

Tags: algebra , CMO

Anya and Vanya’s houses are located on the straight road. The distance between their houses is divided by a shop and a school into three equal parts. If Anya and Vanya leave their houses at the same time and walk towards each other, they will meet near the shop. If Anya rides a scooter, then her speed will increase by $150\,\text{m/min}$, and they will meet near the school. Find Vanya’s speed of walking.

2023 Canada National Olympiad, 3

Tags: geometry , CMO

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: $$\frac{AH}{DX}+\frac{BH}{EY}+\frac{CH}{FZ} \geq 3.$$

2021 Caucasus Mathematical Olympiad, 5

Tags: number theory , CMO

Let $a, b, c$ be positive integers such that the product $$\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a) $$ is a perfect square. Prove that the product $$\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a) $$ is also a perfect square.

2021 Canada National Olympiad, 2

Tags: algebra , inequalities , CMO , Canada , n-variable inequality

Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

2025 Caucasus Mathematical Olympiad, 4

Tags: geometry , CMO

In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal, and they intersect at $E$. Perpendicular bisectors of $AB$ and $CD$ intersect at point $P$ lying inside triangle $AED$, and perpendicular bisectors of $BC$ and $DA$ intersect at point $Q$ lying inside triangle $CED$. Prove that $\angle PEQ = 90^\circ$.

2014 Canada National Olympiad, 2

Tags: floor function , combinatorics proposed , combinatorics , CMO

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

2023 Canada National Olympiad, 2

Tags: combinatorics , CMO

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 $A$.)

2025 Caucasus Mathematical Olympiad, 1

Tags: number theory , CMO

For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$ [list=a] [*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation. [*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation. [/list] [i](Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)[/i]

2023 Canadian Junior Mathematical Olympiad, 4

Tags: combinatorics , CMO

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 $A$.)