There are many hurdles to jump upon before you appear for the IMO.Preparation for it can be done in various ways like:

1. Previous year's question papers for IMO,IN...

The 4th answer is pretty obvious on putting values, we get n = 44. Hence, largest prime factor=11. The 3rd answer is also pretty easy....

Let the polynomial be a + bx + cx^2 + dx^3 + .......

Now, since the polynomial is divisible by n for all ...

I read about it , you are correct. I was taught this method in school so I assumed that it would be correct.I don't know why we are not taught the whole thing in school. In class x , they teach you something and then in class x + y , they contradict it. Its so disgusting!

The problem with your solution is that the general form of Pythagorean triplets is not (2m, m^2 + 1, m^2-1)- it's actually (2mn, m^2 + n^2, m^2 - n^2). You cannot assume that n = 1.