IMO 2020 Shortlist C5
Let p be an odd prime, and put N “ 1 pp3 ´ pq ´ 1. The numbers 1,2,...,N are painted arbitrarily in two colors, red and ...
Category: Combinatorics
Problem
Let p be an odd prime, and put N “ 1 pp3 ´ pq ´ 1. The numbers 1,2,...,N are painted arbitrarily in two colors, red and blue. For any positive integer n ď N, denote by rpnq the fraction of integers in t1,2,...,nu that are red. Prove that there exists a positive integer a P t1,2,...,p ´ 1u such that rpnq ‰ a{p for all n “ 1,2,...,N. (Netherlands)