2019 IMO Shortlist A2

Let $u_{1}, u_{2}, \ldots, u_{2019}$ be real numbers satisfying $$u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 .$$ 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 $$a b \leq-\frac{1}{2019}$$ (Germany)

令 $u_{1}, u_{2}, \ldots, u_{2019}$ 为满足 $$u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { 和 } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 的实数。$$ 设 $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ 和 $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$。证明 $$a b \leq-\frac{1}{2019}$$ (德国)