Problem

Source: 2021 Ukraine NMO 10.7

Tags: Sequence, algebra



The sequence a1,a2,...,a2n of integers is such that each number occurs in no more than n times. Prove that there are two strictly increasing sequences of indices b1,b2,...,bn and c1,c2,...,cn are such that every positive integer from the set {1,2,...,2n} occurs exactly in one of these two sequences, and for each 1in is true the condition abiaci . (Anton Trygub)