# 2020-22 Regular simplex

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 $$n$$).

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