YLG_123 wrote: Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$ Because $$\sum_{i=1}^n\frac{1}{1-x_i}\sum_{1\leq i<j\leq n}x_ix_j=\sum_{k=1}^n\frac{x_k\sum\limits_{i\neq k}x_i+\sum\limits_{1\leq i<j\leq n,i\neq k,j\neq k}x_ix_j}{\sum\limits_{i\neq k}x_k}=$$$$=1+\sum_{k=1}^n\frac{\sum\limits_{1\leq i<j\leq n,i\neq k,j\neq k}x_ix_j}{\sum\limits_{i\neq k}x_k}\leq1+\sum_{k=1}^n\frac{\binom{n-1}{2}}{\binom{n-1}{1}}x_k=1+\frac{\binom{n-1}{2}}{\binom{n-1}{1}}=\frac{n}{2}.$$

YLG_123 wrote: Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$ China Western Mathematical Olympiad 2015