Question 12
Let $p$ be an odd prime and let $N = \frac{1}{4}(p^3 - p) - 1$. The numbers $1, 2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \le N$, denote by $r(n)$ the fraction of integers in $\{1, 2, \dots, n\}$ that are red (number of red numbers divided by $n$). Prove that there exists a positive integer $a \in \{1, 2, \dots, p-1\}$ such that $r(n) \ne \frac{a}{p}$ for all $n = 1, 2, \dots, N$