Problem

Source: China TST 2007, Problem 5

Tags: Geometry inequality



Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A=\left |\sum^n_{i=1}x_i\right |\not =0$ and $ B=\max_{1\leq i<j\leq n}|x_j-x_i|\not =0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that \[ \left |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\alpha_i|.\]