Question 75
Let $n$ be a positive integer and $a_1 \le a_2 \le \cdots \le a_n$ be positive real numbers. Show that
$$
\left( \sum_{k=1}^{n} a_k^2 \right) \left( \sum_{k=1}^{n} k a_k \right) \le \left( \sum_{k=1}^{n} a_k \right) \left( \sum_{k=1}^{n} k a_k^2 \right).
$$
$$
\left( \sum_{k=1}^{n} a_k^2 \right) \left( \sum_{k=1}^{n} k a_k \right) \le \left( \sum_{k=1}^{n} a_k \right) \left( \sum_{k=1}^{n} k a_k^2 \right).
$$