2020 IMO Shortlist C5

Let $p$ be an odd prime, and put $N=\frac{1}{4}\left(p^{3}-p\right)-1$. The numbers $1,2, \ldots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leq N$, denote by $r(n)$ the fraction of integers in $\{1,2, \ldots, n\}$ that are red. Prove that there exists a positive integer $a \in\{1,2, \ldots, p-1\}$ such that $r(n) \neq a / p$ for all $n=1,2, \ldots, N$. (Netherlands)

设$p$为奇素数，并设$N=\frac{1}{4}\left(p^{3}-p\right)-1$。数字 $1,2, \ldots, N$ 被任意涂成两种颜色：红色和蓝色。对于任何正整数 $n \leq N$，用 $r(n)$ 表示 $\{1,2, \ldots, n\}$ 中红色整数的分数。证明存在一个正整数 $a \in\{1,2, \ldots, p-1\}$ 使得 $r(n) \neq a / p$ 对于所有 $n=1,2, \ldots, N$。 (荷兰)