Question 96
Let $n$ be a positive integer, and let $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n}$ be positive real numbers such that $x_{1}+\ldots+x_{n}=y_{1}+\ldots+y_{n}=1$. Show that
$$
\left|x_{1}-y_{1}\right|+\ldots+\left|x_{n}-y_{n}\right| \leq 2-\min_{1 \leq i \leq n} \frac{x_{i}}{y_{i}}-\min_{1 \leq i \leq n} \frac{y_{i}}{x_{i}}
$$
$$
\left|x_{1}-y_{1}\right|+\ldots+\left|x_{n}-y_{n}\right| \leq 2-\min_{1 \leq i \leq n} \frac{x_{i}}{y_{i}}-\min_{1 \leq i \leq n} \frac{y_{i}}{x_{i}}
$$