Question 6
Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers.
Prove that
\[ \sum_{i, j = 1}^{n} \min\{a_i a_j, b_i b_j\}
\le \sum_{i, j = 1}^{n} \min\{a_i b_j, a_j b_i\}. \]
Prove that
\[ \sum_{i, j = 1}^{n} \min\{a_i a_j, b_i b_j\}
\le \sum_{i, j = 1}^{n} \min\{a_i b_j, a_j b_i\}. \]