MathDB

3

Part of 2023 Serbia National Math Olympiad

Problems(1)

Inequality on a periodic sequence

Source: Serbia MO 2023 P3

4/2/2023
Given are positive integers m,nm, n and a sequence a1,a2,…,a_1, a_2, \ldots, such that ai=ai−na_i=a_{i-n} for all i>ni>n. For all 1≤j≤n1 \leq j \leq n, let ljl_j be the smallest positive integer such that m∣aj+aj+1+…+aj+lj−1m \mid a_j+a_{j+1}+\ldots+a_{j+l_j-1}. Prove that l1+l2+…+ln≤mnl_1+l_2+\ldots+l_n \leq mn.
number theory