Number theory

if $a + b = k$ , let $S$ denote the set of all $a, b$ pairs. $S = {(a, b) \in R^{2} | a + b = k}$ . Its a fun exercise to prove this.

max_{(a, b) \in S} a \cdot b = \frac{k^{2}}{4}

where

\arg max_{(a, b) \in S} a \cdot b = (\frac{k}{2}, \frac{k}{2})

For example, if a+b = 10, the maximum of ab = 25, where a = 5, b=5