Question 19
Find all integers $n=2k+1\gt1$ for which there exists a permutation $a_0,a_1,\dots,a_k$ of $\{0,1,\dots,k\}$ satisfying
$$
a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv\cdots\equiv a_k^2-a_{k-1}^2\pmod n.
$$
$$
a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv\cdots\equiv a_k^2-a_{k-1}^2\pmod n.
$$