P5

Part of 2022 Serbia National Math Olympiad

Problems(1)

Game with numbers

Source: Serbian national olympiad 2022 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}

combinatoricsSerbian competitiongameCombinatorics processes