CN-TST 2014 Problem 16

Let $k$ be a fixed odd integer, $k>3$ . Prove: There exist infinitely many positive integers $n$ , such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$ , and $d_1+d_2=n+k$ .