Question 10
Let $n \geqslant 3$ be an integer. An integer $m \geqslant n+1$ 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.
Prove that there are only finitely many positive integers which are not $n$-colourful. Find the largest among them.