AIME I 2009 #8

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] Consider all possible positive differences of pairs of elements of [unparseable or potentially dangerous latex formula] Let [unparseable or potentially dangerous latex formula] be the sum of all of these differences. Find the remainder when [unparseable or potentially dangerous latex formula] is divided by [unparseable or potentially dangerous latex formula]

[SumitG]

For lack of knowledge on how to write sums, I'm going to say for [unparseable or potentially dangerous latex formula],
[unparseable or potentially dangerous latex formula]
We see that in the expanded form of the sum [unparseable or potentially dangerous latex formula] is summed [unparseable or potentially dangerous latex formula] times, where as each power of [unparseable or potentially dangerous latex formula] lower than [unparseable or potentially dangerous latex formula] is summed [unparseable or potentially dangerous latex formula] times, where [unparseable or potentially dangerous latex formula] is the amount ten minus the power that is being summed.
After we notice this, we can simplify our sum to [unparseable or potentially dangerous latex formula].
Then to find the remainder of [unparseable or potentially dangerous latex formula] when divided by [unparseable or potentially dangerous latex formula], we can simply take [unparseable or potentially dangerous latex formula], and we see that our answer is [unparseable or potentially dangerous latex formula]