MathNet 2018-part-05 — 181 problems (EN–VI) Question 68

Find all positive integers $k$ such that there exists some permutation of $(1,2, \ldots, 1000)$, namely $\left(a_{1}, a_{2}, \ldots, a_{1000}\right)$, satisfying $\left|a_{i}-i\right|= k$ for all $i=1,2,\ldots,1000$.