IMC 2015 · Problem 5

Let $n \ge 2$, let $A_1, A_2, \ldots, A_{n+1}$ be $n + 1$ points in the $n$-dimensional Euclidean space, not lying on the same hyperplane, and let $B$ be a point strictly inside the convex hull of $A_1, A_2, \ldots, A_{n+1}$. Prove that $\angle A_iBA_j > 90^\circ$ holds for at least $n$ pairs $(i, j)$ with $1 \le i < j \le n + 1$.