font style="font-weight: bold;">Two positive integers differ by 
I got rid of one of the conditions because it just made more busy work, but it's really still the same problem, just with a different answer.
Call the integers x and x + 60. Then:
At this point, it was rather intuitive to me that $x(x+60)$ must be a perfect square for this to be possible, since I had already done a lot of work with adding radicals. usamts??? I've said too much ;) . But you can show this by squaring both sides of the equation.
Now let z = x+30:
I'm telling you, that difference of squares is key. One of the 2006 IMO problems was a tough looking number theory problem - until you see the difference of squares. Even Alex Zhai didn't see it immediately! And he's like...omnipotent, when it comes to math. It's like he's THERE when they're writing the problems. Just goes to show you that you can never overlook simple tools.
Since we will maximize x by maximizing z, we'll go through the cases, starting with where z is the largest:
Solving for z yields a number that is not an integer, so we throw it out.
Solving for z yields z = 226, so x = 196 .Thus, our desired sum is:
If you're curious, the condition that I excluded was that our root sum cannot be an integer, which this answer doesn't fit. I took it out because I just would've had to type up another case, and I'm sure you got the idea.
AIME - Number Theory
"Dying is the day worth living for"
So I finally started watching At World's End. I don't know why everybody said it was awful. I mean, granted, it was no Dark Knight (which was absolutely mind blowing), but it was worth a watch.
I also started watching a new anime called Monster. The premise is that this doctor saved the life of a small boy who grew up to be a serial killer. So the killer in a way idealizes the doctor, while the doctor is trying to track him down. It reminded me of Death Note a bit, only without the magic notebooks or flying shinigami.
I'm also officially enrolled in WOOT now. I'm wondering if it'll be enough to keep me competitive. After all, all of the MOSP participants automatically get into it. Then again, I think I'm progressing rather quickly. WOOT should help me out with that. We'll see.
Made up number theory problem
Prove that, for all positive integers x and r, the remainder when 
This is pretty easy if you interpret it as:
Which, according to the binomial theorem, is:
If we expand this out, then every term except for the first will be divisble by (x-1). The first term is just 1, so the remainder is 1. Nifty, right? I thought so.
WOOT
No, that is NOT the douchalicious phrase that people say online when something causes them to orgasm all over their keyboard. Sorry, I'm a bit angry right now. Anyway, it stands for Worldwide Olympiad Online Training. And I've managed to convince one of my parents (guess which one) to sign me up for it. I'm REALLY happy, even if you can't tell from my constant unintentionally unsatisfied mood. It's basically a USAMO prep class.
I've been wondering, would it be possible for me, somebody who had never heard of the AMC less than a year ago, to make it to the IMO? It's certainly a HUGE stretch. I was looking at the winning USAMO scores from this year, they were all in the 33 - 37 range (two people scored 37, the highest). That would mean that I would probably have to solve EVERY problem at least somewhat. Several of them would have to be perfect. I guess I just have to ask myself one thing: Do I think I have what it takes to practice, learn what needs to be learned, and, when the time comes, shit in the mouth of 6 math problems? Of course I can. I'm going for it.
Wish me luck.