Spring 2017, problem 47
Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once, show that there exist integers $\ell $ and $m,\ 1\lt\ell \lt m$, such that $a_1+a_m=2a_{\ell}$.
Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once, show that there exist integers $\ell $ and $m,\ 1\lt\ell \lt m$, such that $a_1+a_m=2a_{\ell}$.