Question 26
(CZS 1) Let $n\ge2$ be a natural number and let the real numbers $p,a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ satisfy $1/2\le p\le1$, $0\le a_i$, $0\le b_i\le p$, $i=1,\ldots,n$, and $\sum_{i=1}^n a_i=\sum_{i=1}^n b_i=1$. Prove the inequality
$$\sum_{i=1}^n b_i\prod_{\substack{j=1\\j\ne i}}^n a_j\le\frac p{(n-1)^{n-1}}.$$
$$\sum_{i=1}^n b_i\prod_{\substack{j=1\\j\ne i}}^n a_j\le\frac p{(n-1)^{n-1}}.$$