@VibhavPant i someone has a background in handling Olympiad questions and knows a little bit of real analysis and calculus .. the paper is not at all difficult :-)
@Chris'ssis from $(11)$ on it is just some simple manipulations to get things looking like a simple sum... You have a one line proof? I don't doubt there is one.
Quite well, thanks :) Played tennis at 7:45 AM before the heat became too horrid ... Now trying to construct an answer to a question here I used to know how to do when I was a graduate student.
@Hippa: I take it you saw my response to your problem after you went to bed. I will bet you thousands of dollars there's no way to solve explicitly for those curves as graphs $y=y(x)$.
@Chris'ssis Both you and gar mention the recurrence with $I_k$. It would be nice to see it justified (I know it is routine, but it is necessary at some point).
@Chris'ssis certainly I could have compressed my proof to a few lines if I wanted to leave out some intermediate steps, but I like to have it so that it is easy to follow the proof.
@Chris'ssis and the recursion with $I_k$ is certainly nice, but it has been used three times (if you count Ian's continuation of gar's answer) but never been actually shown. Perhaps I will add it to my answer so that no one with a short answer need sully theirs. :-)
I hate how certain proofs can be so rigirious about obvious things, and suddenly state something out of no where that is the most essential thing to the proof without even explaining it..
@Ted if everyone would've done that.. I've seen so many proofs explaining the pignehole principle all over again, only to then state something like "obviously the path is of even length" which is so crucial to the proof, without explaining why..
Oh, it changed. They're both excellent, @skull. MIT has a very strong applied group, Harvard less so. They're both ranked in the top 5 departments in the country.
@Pedro: I just saw a pun that is in your style on my Facebook page: When attacked by a mob of clowns, go for the juggler :D
