AIME - Complex Numbers / Probability

Today everyone was on an NHS field trip (maybe later I'll explain why I'm not in NHS...) and so in most of our classes we didn't really do anything. So I brought in a bunch of old AIME problems to work on. This one caught my attention while my physics teacher started giving a random lecture on economics:



Some of those words are blue because I copied that from the AoPS wiki.

Speaking of the AoPS wiki, I hate some of the solutions they have on there. Mainly because they're hard to follow. I suppose they're just supposed to be general solution outlines, but the one for this was some odd use of a bunch of trig formulas, and it wasn't very nice to look at. I like my solution because once it's explained, the problem becomes rather intuitive.

I interpretted this problem geometrically. It makes it a lot easier to see what's going on. So we have some roots of unity. At this point, I didn't even consider the number 1997. Then we're adding another root of unity to it. If we treat the complex numbers as vectors, then this is just the head-to-tail method. The diagram shows some of the possibilities of this. The dashed circles are all possible roots of unity. I just drew in two possibilities:



The absolute value of this number, which is just its distance from the origin, is greater than that odd square root. In other words, it's outside the circle of that radius. We can sketch that in too to get a better look:



Hmm. If we can visualize sliding that second "unity" circle around the first, the same percentage of it should always be outside that circle that we drew in. That suggests that the probability is independent of the first choice, which certainly makes things easier.

One issue that's worth addressing at this point is whether our approach of throwing away the number 1997 for simplicity has thrown us off course. The same percentage of the circle is always farther from the origin than that silly circle with the square root radius, but we're not looking at the percentage of the circle, we're looking at the number of roots of unity for the number 1997 that lay outside the circle. If we slide that second circle around, there's no guarentee that this will be the same for each. When I did the problem out, I reasoned that it is, because if you draw in all the actual possibilities for a certain number, you can rotate the entire thing by 360 / n degrees, causing each set of possibilties to translate to the next exactly...if that makes sense. I hope it does T_T.

Anyway, we can make this problem easier by only considering the case where u = 1. Since the number of roots outside of that larger circle should be the same for each case, then the probability for all cases will be the same as the probability for one case. Here's an altered diagram for this case:



If we can figure out where they intersect, then we can figure out the range of that arc that is outside the circle, and figure out how many roots lay on it! The equation of each circle is:



We can expand the second equation and substitute the first into it to get:



That familiar sqrt(3)/2 expression is encouraging. What about the +1? Remember that our circle is translated over to the right by +1. So if we consider the center of that circle the origin, then they intersect at x = sqrt(3)/2 = cos(+/- pi/6). Aha, now we have a range! If we let our angle be:



(this is the general angle of a 1997th root of unity)

Then it lies within the range:



Since the floor of 1997 / 12 is 166, then there are 166 positive integers and 166 negative integers in this range. We aren't counting k=0 because this wouldn't be a distinct root. Therefore, we have 332 total roots in this range. We are selecting them from 1996 roots (remember, that root we already picked is off limits), therefore, the probability is:



And our final answer is 499 + 83 = 582.

The lesson learned? When doing a problem with complex numbers, take some time to consider which way it's more profitable to interpret it: algebraically, or geometrically, as complex numbers are essentially a mix between these two topics.


OH. So why aren't I in NHS? Because it's a cult where all they do is struggle to sell crap candy and do random community service for NHS points, the currency of NHS. You gain nothing from joining, besides maybe having an extra activity on your college resume. You lose your soul. Not a great trade in my oppinion. Also, they appreciate kids who stack on the L4 classes more than the kids who actually focus on subject and become really good at it. And their advisor is a stubborn old man too, according to most of my friends in NHS.