Olympiad problems

2024 Germany Team Selection Test, 1

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

2024 Azerbaijan BMO TST, 1

Tags: IMO Shortlist , number theory , AZE BMO TST , AZE EGMO TST , TST , BRA Cono Sur TST

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

2023 Azerbaijan BMO TST, 3

Tags: algebra , functional equation , AZE BMO TST , AZE EGMO TST , BMO Shortlist , TST

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2024 Azerbaijan BMO TST, 4

Tags: combinatorics , AZE BMO TST , TST

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

2023 Balkan MO Shortlist, G5

Tags: geometry , AZE BMO TST , TST , Miquel point

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

2022 Balkan MO Shortlist, A4

Tags: algebra , functional equation , AZE BMO TST , AZE EGMO TST , BMO Shortlist , TST

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2024 Brazil Cono Sur TST, 1

Tags: IMO Shortlist , number theory , AZE BMO TST , AZE EGMO TST , TST , BRA Cono Sur TST

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.