MathDB

Let

u_1, u_2, \dots, u_{2019}

be real numbers satisfying u_{1}+u_{2}+\cdots+u_{2019}=0 \text { and } u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1. Let

a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)

and

b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)

. Prove that

a b \leqslant-\frac{1}{2019}.

Problem Statement