Q25 Matemática  (IMO Shortlist 1991)

Seja $n\ge 2,n\in \mathbb{N}$ e $p,a_1,a_2,\dots ,a_n,b_1,b_2,\dots ,b_n\in \mathbb{R}$ satisfazendo $\frac{1}{2}\le p\le 1,$ $0\le a_i,$ $0\le b_i\le p,$ $i=1,\dots ,n,$ e $$\sum _{i=1}^n{a}_i=\sum _{i=1}^n{b}_i.$$ Prove a desigualdade: $$\sum _{i=1}^n{b}_i\prod _{j=1,j\ne i}^n{a}_j\le \frac{p}{(n-1{)}^{n-1}}.$$