2021 IMO Shortlist Problems/C2

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Problem

Let $n\ge 3$ be an integer. An integer $m\ge n-1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_1, C_2,\dots, C_n$, it is possible to place $m$ of them around a circle so that in any group of $n+1$ consecutive marbles there is at least one marble of colour $C_i$ for each $i = 1,\dots, n$. Prove that there are only finitely many positive integers which are not $n$-colourful. Find the largest among them.

Solution

https://youtu.be/Uw0G8uLDc2I