Question 13
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 $n \leqslant N$, 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 $n=1,2, \ldots, N$.
Prove that there exists a positive integer $a \in\{1,2, \ldots, p-1\}$ such that $r(n) \neq a / p$ for all $n=1,2, \ldots, N$.