Olympiad problems

2025 Euler Olympiad, Round 2, 5

We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations: [b]1. [/b]Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between. [b]2. [/b]Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right. [b]3.[/b] Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells. The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player. [img]https://i.imgur.com/IjcIDOa.png[/img] [i]Proposed by Luka Tsulaia, Georgia[/i]

2025 Euler Olympiad, Round 1, 9

Tags: geometry , Euler Olympiad

Three circles with radii $1$, $2$, and $3$ are pairwise tangent to each other. Find the radius of the circle that is externally tangent to all three of these circles. [i]Proposed by Tamar Turashvili, Georgia [/i]

2025 Euler Olympiad, Round 1, 5

Tags: Euler Olympiad , number theory

Find the minimum value of $m + n$, where $m$ and $n$ are positive integers satisfying: $2023 \vert m + 2025n$ $2025 \vert m + 2023n$ [i]Proposed by Prudencio Guerrero Fernández [/i]

2025 Euler Olympiad, Round 1, 6

Tags: Euler Olympiad , geometry , ratio

There are seven rays emanating from a point $A$ on a plane, such that the angle between the two consecutive rays is $30 ^{\circ}$. A point $A_1$ is located on the first ray. The projection of $A_1$ onto the second ray is denoted as $A_2$. Similarly, the projection of $A_2$ onto the third ray is $A_3$, and this process continues until the projection of $A_6$ onto the seventh ray is $A_7$. Find the ratio $\frac{A_7A}{A_1A}$. [img]https://i.imgur.com/oxixe5q.png[/img] [i]Proposed by Giorgi Arabidze, Georgia[/i]

2025 Euler Olympiad, Round 1, 2

Tags: Euler Olympiad , algebra

Find all five-digit numbers that satisfy the following conditions: 1. The number is a palindrome. 2. The middle digit is twice the value of the first digit. 3. The number is a perfect square. [i]Proposed by Tamar Turashvili, Georgia [/i]

2025 Euler Olympiad, Round 2, 2

Tags: Euler Olympiad , geometry , Intersection , circles , midpoint

Points $A$, $B$, $C$, and $D$ lie on a line in that order, and points $E$ and $F$ are located outside the line such that $EA=EB$, $FC=FD$ and $EF \parallel AD$. Let the circumcircles of triangles $ABF$ and $CDE$ intersect at points $P$ and $Q$, and the circumcircles of triangles $ACF$ and $BDE$ intersect at points $M$ and $N$. Prove that the lines $PQ$ and $MN$ pass through the midpoint of segment $EF$. [i] Proposed by Giorgi Arabidze, Georgia[/i]

2025 Euler Olympiad, Round 1, 7

Tags: Euler Olympiad , algebra

Let $s(n)$ be the final value obtained after repeatedly summing the digits of $n$ until a single-digit number is reached. (For example: $s(187) = 7$, because the digit sum of $187$ is $16$ and the digit sum of $16$ is $7$). Evaluate the sum: $$ s(1^2) + s(2^2) + s(3^2) + \ldots + s(2025^2)$$ [i]Proposed by Lia Chitishvili, Georgia [/i]

2025 Euler Olympiad, Round 2, 4

Tags: Euler Olympiad , functional equation , Cauchy functional equation , multiplicative function , algebra

Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$, $$ f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y), $$ where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$. [I]Proposed by Stijn Cambie, Belgium[/i]

2024 Euler Olympiad, Round 1, 3

Tags: geometry , Euler Olympiad

In a convex trapezoid $ABCD$, side $AD$ is twice the length of the other sides. Let $E$ and $F$ be points on segments $AC$ and $BD$, respectively, such that $\angle BEC = 70^\circ$ and $\angle BFC = 80^\circ$. Determine the ratio of the areas of quadrilaterals $BEFC$ and $ABCD$. [i]Proposed by Zaza Meliqidze, Georgia [/i]

2025 Euler Olympiad, Round 1, 10

Tags: Euler Olympiad , combinatorics , counting

There are 12 gold stars arranged in a circle on a blue background. Giorgi wants to label each star with one of the letters $G$, $E$, or $O$, such that no two consecutive stars have the same letter. Determine the number of distinct ways Giorgi can label the stars. [img]https://i.imgur.com/qIxdJ8j.jpeg[/img] [i]Proposed by Giorgi Arabidze, Georgia [/i]

2025 Euler Olympiad, Round 1, 3

Tags: Euler Olympiad , algebra

Evaluate the following sum: $$ \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \frac{1}{1 + 2 + 3 + 4} + \ldots + \frac{1}{1 + 2 + 3 + 4 + \dots + 2025} $$ [i]Proposed by Prudencio Guerrero Fernández[/i]

2025 Euler Olympiad, Round 1, 4

Tags: Euler Olympiad , geometry , ratio

Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$ Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized. [img]https://i.imgur.com/NfjRpgP.png[/img] [i] Proposed by Andria Gvaramia, Georgia [/i]

2025 Euler Olympiad, Round 1, 8

Tags: ratio , algebra , Euler Olympiad

Let $S$ be the set of non-negative integer powers of $3$ and $5$, $S = \{1, 3, 5, 3^2, 5^2, \ldots \}$. For every $a$ and $b$ in $S$ satisfying $$ \left| \pi - \frac{a}{b} \right| < 0.1 $$ Find the minimum value of $ab$. [i]Proposed by Irakli Shalibashvili, Georgia [/i]

2025 Euler Olympiad, Round 2, 3

Tags: Euler Olympiad , algebra , functional equation , trigonometry

Find all functions $ f : \mathbb{R} \to \mathbb{R} $ such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

2025 Euler Olympiad, Round 1, 1

Tags: algebra , Euler Olympiad

Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board. [i]Proposed by Giorgi Arabidze, Georgia[/i]

2025 Euler Olympiad, Round 2, 6

Tags: Euler Olympiad , algebra , combinatorics , number theory , graph theory , Iteration

For any subset $S \subseteq \mathbb{Z}^+$, a function $f : S \to S$ is called [i]interesting[/i] if the following two conditions hold: [b]1.[/b] There is no element $a \in S$ such that $f(a) = a$. [b]2.[/b] For every $a \in S$, we have $f^{f(a) + 1}(a) = a$ (where $f^{k}$ denotes the $k$-th iteration of $f$). Prove that: [b]a) [/b]There exist infinitely many interesting functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$. [b]b) [/b]There exist infinitely many positive integers $n$ for which there is no interesting function $$ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}. $$ [i]Proposed by Giorgi Kekenadze, Georgia[/i]

2025 Euler Olympiad, Round 2, 1

Tags: Euler Olympiad , number theory , Divisors , relatively prime , prime numbers

Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]