Problem

Source: Romania TST 2016 Day 2 P3

Tags: number theory, algebra



Prove that: (a) If (an)n1 is a strictly increasing sequence of positive integers such that a2n1+a2nan is a constant as n runs through all positive integers, then this constant is an integer greater than or equal to 4; and (b) Given an integer N4, there exists a strictly increasing sequene (an)n1 of positive integers such that a2n1+a2nan=N for all indices n.