Olympiad problems

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2023 Pan-American Girls’ Mathematical Olympiad, 1

An integer $n \geq 2$ is said to be [i]tuanis[/i] if, when you add the smallest prime divisor of $n$ and the largest prime divisor of $n$ (these divisors can be the same), you obtain an odd result. Calculate the sum of all [i]tuanis[/i] numbers that are less or equal to $2023$.

2018 Finnish National High School Mathematics Comp, 4

Tags: geometric transformation , geometry , number theory , largest prime divisor , game , combinatorics , game strategy

Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$. Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.