IR-TST 2014 Problem 17

Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$ , call a real number $a$ good if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$ . Also a subset of $X$ is called compact if the average of its elements is a good number. Prove that at least $2^{n-3}$ subsets of $X$ are compact.

Proposed by Mahyar Sefidgaran