IMC 2009 · Problem 5

Let $n$ be a positive integer. An $n$-simplex in $\mathbb{R}^n$ is given by $n+1$ points $P_0, P_1, \dots, P_n$, called its vertices, which do not all belong to the same hyperplane. For every $n$-simplex $S$ we denote by $v(S)$ the volume of $S$, and we write $C(S)$ for the center of the unique sphere containing all the vertices of $S$.

Suppose that $P$ is a point inside an $n$-simplex $S$. Let $S_i$ be the $n$-simplex obtained from $S$ by replacing its $i$-th vertex by $P$. Prove that

$$v(S_0)C(S_0) + v(S_1)C(S_1) + \dots + v(S_n)C(S_n) = v(S)C(S).$$