Problem - 3230
(Rearrangement Theorem) Let $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ be sequences of positive real numbers, and let $c_1, c_2, \cdots, c_n$ be a permutation of $b_1, b_2, \cdots, b_n$. The sum $S=a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximal if the two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are sorted in the same way and minimal if the two sequences are sorted oppositely, one increasing and the other decreasing.
Let $a_i > a_j$. Consider the sums
\begin{align}
S_1 &= a_1c_1 + \cdots + a_ic_i + \cdots + a_jc_j + \cdots + a_nc_n\\
S_2 &= a_1c_1 + \cdots + a_ic_j + \cdots + a_jc_i + \cdots + a_nc_n
\end{align}
We have obtained $S_2$ from $S_1$ by switching the positions of $c_i$ and $c_j$. Then
$$S_1-S_2 = (a_i-a_j)(c_i-c_j)$$
Therefore $c_i>c_j \implies S_1 > S_2$ and $c_i < c_j \implies S_1 < S_2$.