### International Mathematics Competition for University Students

July 27 - Aug 2 2015, Blagoevgrad, Bulgaria

Home

Results
Individuals
Teams

Official IMC site

### Problem 5

5. Let $n\ge2$, 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 $\displaystyle{1\le i<j\le n+1}$.

Proposed by Géza Kós, Eötvös University, Budapest