OBM 2022 Problem 5

Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$ , $S(n) \geq n$ , and such that there are not positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$ . Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$ , $S(n) \geq cn^{\frac{3}{2}}$ .