MathDB

Let

p

be an odd prime, and put

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 \leqslant N,

denote

r(n)

the fraction of integers

\{ 1,2, \dots, n \}

that are red. Prove that there exists a positive integer

a \in \{ 1,2, \dots, p-1\}

such that

r(n) \neq a/p

for all

n = 1,2, \dots , N.

[I]Netherlands

Problem Statement