Question 67
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
$$
a b \leqslant-\frac{1}{2019}$$
$$
u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text{and} \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 .$$
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
$$
a b \leqslant-\frac{1}{2019}$$