KURSCHAK 2008 Problem 2

Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$ , and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$ . (Here, the powers of $2$ are $1,2,4,\dots$ .) Prove that $|S|\le |T|$ .