AIME - Number Theory

font style="font-weight: bold;">Two positive integers differ by The sum of their square roots is the square root of an integer. What is the maximum possible sum of the two integers?

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.