Balancing the reaction of potassium permanganate and sulfuric acid with hydrogen peroxide
It seems to me that this is simply a redox reaction in the presence of acid.  

If you write the 2 half reactions, and balance in the presence of acid (sulfuric acid), you end up with the same answer (with 5 O2 instead of 10 O on the product side).  

This seems to be more of a mathematical manipulation rather than a chemistry problem, IMHO.