Let $S$ be the unit sphere in $\mathbb{R}^n$, centered at the origin, and $P_1 P_2 \dots P_{n+1}$ a regular simplex inscribed in $S$. Prove that for a point $P$ inside $S$,
$$\sum_{i=1}^{n+1} (PP_i)^4$$
depends only on the distance $OP$ (and $n$).

The best solution was submitted by 이준호 (수리과학과 2016학번, +4). Congratulations!

Here is his solution of problem 2020-22.

Other solutions was submitted by 고성훈 (수리과학과 2018학번, +3), 채지석 (수리과학과 2016학번, +3).