2000 AIME I Problems/Problem 6

Problem

For how many ordered pairs of integers is it true that and that the arithmetic mean of and is exactly more than the geometric mean of and ?

Solution

From the condition given,

$\begin{align*} \frac{x + y}{2} - 2 &= \sqrt{xy}\\ x + y - 2\sqrt{xy} &= 4\\ (\sqrt{y} - \sqrt{x})^2 &= 4\\ \sqrt{y} - \sqrt{x} &= 2 \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

The last equation is true because .

Here, we can count how many valid pairs of satisfy our equation, rather than directly, because can get messy.

The maximum that can be is because must be an integer (this is because , an integer). Then , and we continue this downward until , in which case . The number of pairs of , and so is, then, .