Question 100
Let $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ be positive real numbers so that
$$
x_1 + x_2 + \dots + x_n \geq x_1 y_1 + x_2 y_2 + \dots + x_n y_n.
$$
Show that for any non-negative integer $p$ the following inequality holds
$$
\frac{x_1}{y_1^p} + \frac{x_2}{y_2^p} + \dots + \frac{x_n}{y_n^p} \geq x_1 + x_2 + \dots + x_n.
$$
$$
x_1 + x_2 + \dots + x_n \geq x_1 y_1 + x_2 y_2 + \dots + x_n y_n.
$$
Show that for any non-negative integer $p$ the following inequality holds
$$
\frac{x_1}{y_1^p} + \frac{x_2}{y_2^p} + \dots + \frac{x_n}{y_n^p} \geq x_1 + x_2 + \dots + x_n.
$$