AIME II #9 2008

[stupidityismygam]

<my favorite problem>

A particle is located on the coordinate plane at [unparseable or potentially dangerous latex formula] Define a [unparseable or potentially dangerous latex formula] for the particle as a counterclockwise rotation of [unparseable or potentially dangerous latex formula] radians about the origin followed by a translation of [unparseable or potentially dangerous latex formula] units in the positive x-direction. Given that the particle's position after [unparseable or potentially dangerous latex formula] moves is [unparseable or potentially dangerous latex formula] find the greatest integer less than or equal to [unparseable or potentially dangerous latex formula]

[Michael T]

I got fifteen, and I have no idea how to get 19.
Somehow I determined every four moves returns back to (5,0), which has to be wrong considering the answer...
Help?

[collegebookworm]

[stupidityismygam]

Consider it on the complex plane (which doesn't change the answer, there is just an i on the y-coordinate)

Recall that [unparseable or potentially dangerous latex formula] and the x-coordinate is [unparseable or potentially dangerous latex formula], y-coordinate [unparseable or potentially dangerous latex formula]
Also recall that [unparseable or potentially dangerous latex formula]
Define [unparseable or potentially dangerous latex formula] where the coordinate [unparseable or potentially dangerous latex formula] is the coordinate after [unparseable or potentially dangerous latex formula] moves.
We have that [unparseable or potentially dangerous latex formula]

Each rotation is the same as multiplying [unparseable or potentially dangerous latex formula] and each shift is the same as adding [unparseable or potentially dangerous latex formula]


[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

Adding these looks like a pretty daunting task, but luckily for us [unparseable or potentially dangerous latex formula] has some nice properties.

The one that I will use is that [unparseable or potentially dangerous latex formula]
The proof of this is fairly straightforward. [unparseable or potentially dangerous latex formula] is simply the sum of the [unparseable or potentially dangerous latex formula] roots of unity, which is simply the polynomial [unparseable or potentially dangerous latex formula] By Vieta, the sum of the roots is 0.

So, going back to the problem.
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

So, [unparseable or potentially dangerous latex formula]
Thus the answer is [unparseable or potentially dangerous latex formula](as seen by collegebookworm's calculations)

[collegebookworm]

[stupidityismygam]