IrMO 2013 Problem 9

We say that a doubly infinite sequence $. . . , s_{-2}, s_{-1}, s_{0}, s_1, s_2, . . .$ is subaveraging if $s_n = (s_{n-1} + s_{n+1})/4$ for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such that $s_m = s_n$ , then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .