shortlistBMO 2011

The required maximum is $1/(n-1{)}^n$ and is achieved if and only if the $x_i$ are all equal to $1/(n-1)$. The constraint on the $x_i$ is equivalent to $$\sum _{k=1}^n(k-1){\sigma }_k=1,$$ where $${\sigma }_k=\sum _{1\le i_1<\cdots <i_k\le n}{x}_{i_1}\cdots x_{i_k},k=1,2,\dots ,n.$$ By the AM-GM inequality, $${\sigma }_k\ge \left(\genfrac{}{}{0ex}{}{n}{k}\right){\sigma }_n^{k/n},k=1,2,\dots ,n,$$ so, upon substitution $t={\sigma }_n^{1/n}$, $$\sum _{k=1}^n(k-1)\left(\genfrac{}{}{0ex}{}{n}{k}\right)t^k\le 1;$$ that is, $(t+1{)}^{n-1}((n-1)t-1)\le 0$. Consequently, $t\le 1/(n-1)$. Equality clearly forces all the $x_i$ to be equal to $1/(n-1)$. Since these $x_i$ obey the constraint, the conclusion follows.