Sea $ n \geq 2, n \in \mathbb{N}$ y sean $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ que satisfacen $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i = 1, \ldots, n,$ y \[ \sum^n_{i=1} a_i = \sum^n_{i=1} b_i.\] Demuestra la desigualdad: \[ \sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.\]