AIME I #13 2008

[stupidityismygam]

Let
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]
Suppose that
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]
There is a point[unparseable or potentially dangerous latex formula] for which [unparseable or potentially dangerous latex formula] for all such polynomials, where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are positive integers, [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime, and [unparseable or potentially dangerous latex formula] Find [unparseable or potentially dangerous latex formula]

[collegebookworm]

Well, since it appears no one has done this alone, let's work together.
If we plug in [unparseable or potentially dangerous latex formula], we get [unparseable or potentially dangerous latex formula]; thus we can eliminate that term:
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]
If we plug in [unparseable or potentially dangerous latex formula], we get [unparseable or potentially dangerous latex formula]; plugging in [unparseable or potentially dangerous latex formula] gives [unparseable or potentially dangerous latex formula], and we can add the two equations to get [unparseable or potentially dangerous latex formula]. We can plug that back in to find that [unparseable or potentially dangerous latex formula], thus we have:
[unparseable or potentially dangerous latex formula]
Now we can use the same method with [unparseable or potentially dangerous latex formula] ([unparseable or potentially dangerous latex formula]) and [unparseable or potentially dangerous latex formula] ([unparseable or potentially dangerous latex formula]), and find that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Putting in [unparseable or potentially dangerous latex formula] ([unparseable or potentially dangerous latex formula]) and [unparseable or potentially dangerous latex formula] ([unparseable or potentially dangerous latex formula]), we find that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Now we use [unparseable or potentially dangerous latex formula] to find that [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], and [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

Now comes the speculation:
Perhaps if we try [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], and if we decompose the former term and combine and plug in [unparseable or potentially dangerous latex formula] for x and [unparseable or potentially dangerous latex formula] for y, we get [unparseable or potentially dangerous latex formula].

EDIT: I plugged in [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] and got [unparseable or potentially dangerous latex formula], so perhaps this helps?