2021 IMO Shortlist C2

Let $n \geq 3$ be an integer. An integer $m \geq n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_{1}, C_{2}, \ldots, C_{n}$, 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 $i=1, \ldots, n$. Prove that there are only finitely many positive integers which are not $n$-colourful. Find the largest among them.

设 $n \geq 3$ 为整数。整数 $m \geq n+1$ 称为 $n$ 彩色弹珠，如果给定无限多个具有 $n$ 颜色 $C_{1}、C_{2}、\ldots、C_{n}$ 每种颜色的弹珠，可以将 $m$ 个弹珠放置在一个圆上，以便在任何一组 $n+1$ 连续弹珠中，对于每个 $i=1、\ldots、n$，至少有一个颜色为 $C_{i}$ 的弹珠。证明只有有限多个非 $n$ 色的正整数。找出其中最大的一个。