Q2 Matemática  (IMO Shortlist 2019)

Sejam $u_1,u_2,\dots ,u_{2019}$ números reais que satisfazem $$u_1+u_2+\cdots +u_{2019}=0\text{ e }{u}_1^2+u_2^2+\cdots +u_{2019}^2=1.$$Deixe $a=\min \left(u_1,u_2,\dots ,u_{2019}\right)$ e $b=\max \left(u_1,u_2,\dots ,u_{2019}\right)$. Prove que $$ab⩽-\frac{1}{2019}.$$