AIME I Problem 13 2004

[stupidityismygam]

I like this problem, don't be intimidated by the notation

The polynomial [unparseable or potentially dangerous latex formula] has 34 complex roots of the form [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Given that [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime positive integers, find [unparseable or potentially dangerous latex formula]

[Kurt]

You might be able to use the identity [unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula], but I wouldn't know how to solve for the complex roots of that easily at all....although it does look like it has nice symmetry.

Instead, it's easy to see that [unparseable or potentially dangerous latex formula] is just a geometric sequence, so the sum is:

[unparseable or potentially dangerous latex formula].

And so [unparseable or potentially dangerous latex formula], or:

[unparseable or potentially dangerous latex formula]. Simplifying, we get

[unparseable or potentially dangerous latex formula].

This fortunately factors:
[unparseable or potentially dangerous latex formula].

Thus solutions to [unparseable or potentially dangerous latex formula] are what we are interested in.
[unparseable or potentially dangerous latex formula], so the only solution common to both factors is [unparseable or potentially dangerous latex formula]. However, this solution is excluded because of the denominator in the actual polynomial. Therefore, all of the solutions are distinct.

So all solutions are expansions of [unparseable or potentially dangerous latex formula].

By De Moivre's Theorem, [unparseable or potentially dangerous latex formula]. (And the exact same thing, with 17s, for the other root's case.)

In this problem, all [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula], so the sum of the five lowest [unparseable or potentially dangerous latex formula] is simply the sum of the five lowest fractions with positive integer numerators with 17 or 19 in the denominator.

Since [unparseable or potentially dangerous latex formula] (the five lowest [unparseable or potentially dangerous latex formula]):

[unparseable or potentially dangerous latex formula].

So [unparseable or potentially dangerous latex formula].

[stupidityismygam]

yep, that was my solution too

[zefuri]

yessir althoguh i hate complex numbers