Question 170
Let $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{n}$ be nonnegative real numbers such that $x_{i}+y_{i}=1$ for each $i=1,2, \ldots, n$. Prove that
$$
\left(1-x_{1} x_{2} \ldots x_{n}\right)^{m}+\left(1-y_{1}^{m}\right)\left(1-y_{2}^{m}\right) \ldots\left(1-y_{n}^{m}\right) \geq 1,
$$
where $m$ is an arbitrary positive integer.
$$
\left(1-x_{1} x_{2} \ldots x_{n}\right)^{m}+\left(1-y_{1}^{m}\right)\left(1-y_{2}^{m}\right) \ldots\left(1-y_{n}^{m}\right) \geq 1,
$$
where $m$ is an arbitrary positive integer.