Olympiad problems

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Found problems: 2

2022 Serbia National Math Olympiad, P5

On the board are written $n$ natural numbers, $n\in \mathbb{N}$. In one move it is possible to choose two equal written numbers and increase one by $1$ and decrease the other by $1$. Prove that in this the game cannot be played more than $\frac{n^3}{6}$ moves.

2025 Bulgarian Winter Tournament, 11.3

Tags: combinatorics processes , invariant , combinatorics

We have $ n $ chips that are initially placed on the number line at position 0. On each move, we select a position $ x \in \mathbb{Z} $ where there are at least two chips; we take two of these chips, then place one at $ x-1 $ and the other at $ x+1 $. a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line. b) For every possible final position, let $ \Delta $ represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of $ \Delta $ in terms of $ n $.