AIME #13 1988

[stupidityismygam]

Find [unparseable or potentially dangerous latex formula] if [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are integers such that [unparseable or potentially dangerous latex formula] is a factor of [unparseable or potentially dangerous latex formula]

[kashhustler]

hmm i'm pretty much stumped on trying to find a nice method for this one so i'll just resort to good old brute forcing

basically, we can work backwards to find the coefficients of the other factor of that and then use it to find a and b.

the constant term of the other factor must be -1 because it multiplies with -1 to get 1, then the x term is x, and so on...

i'm too lazy to work it out but hey, not too long of a solution for a #13, probably takes 10-15 minutes

[stupidityismygam]

[zefuri]

add a bracket to the end...well uhhh i cant edit it for you so yeah

[Kurt]

Hmm, well [unparseable or potentially dangerous latex formula] returns the two golden ratio roots, [unparseable or potentially dangerous latex formula]. These also must be roots of the original equation.

I can substitute them in separately and subtract the two equations to get a couple of Binet's formulas to pop out:

Of course, I also would need to divide by [unparseable or potentially dangerous latex formula] for Binet's formula:
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]

So: [unparseable or potentially dangerous latex formula]

Also note [unparseable or potentially dangerous latex formula] (Descartes's Rule of Signs). Opaquely, this implies one integer solution: [unparseable or potentially dangerous latex formula].

[Kurt]

And why are some of these problems stickied?

[stupidityismygam]