AIME I #4 2008

[stupidityismygam]

There exist unique positive integers [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] that satisfy the equation [unparseable or potentially dangerous latex formula]. Find [unparseable or potentially dangerous latex formula].

[Celsion]

Started by completing the square to get:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Next moved the [unparseable or potentially dangerous latex formula] over.
[unparseable or potentially dangerous latex formula]
Used difference of squares to get:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Using a parity argument, we see that both terms have the same parity (both even or both odd). Since 244 is even. It is clear that both must be even.
Factored [unparseable or potentially dangerous latex formula] to get [unparseable or potentially dangerous latex formula].
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

[collegebookworm]

Sorry Celsion, I cheated:
I knew the pythagorean triple pattern for two consecutive evens (once i had completed the square and saw it was an even pythagorean): [unparseable or potentially dangerous latex formula].
[unparseable or potentially dangerous latex formula], because [unparseable or potentially dangerous latex formula], which is halfway between 60 and 62. Since we have [unparseable or potentially dangerous latex formula], I realized that [unparseable or potentially dangerous latex formula], and thus [unparseable or potentially dangerous latex formula].