MathNet 2003 — Part 01 Question 90

Is it true that for any permutation $a_{1}, a_{2}, \ldots, a_{2002}$ of $1,2, \ldots, 2002$ there are positive integers $m$ and $n$ of the same parity such that $1 \leq m\lt n \leq 2002$ and $a_{m}+a_{n}=2 a_{\frac{m+n}{2}}$?