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...

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.

The 6th one can be solved by taking 12 = 2m so the triplets would be (m*m - 1) +( m*m + 1) + 2m

so answer would be 12+ 35 + 37 = 84.