Putnam questions for fun

1. Nov 10, 2006

SeReNiTy

Hey guys, I was doing some Putnam questions for fun and came across something strage to me. In the 2005 Putnam competition, question B3, link provided below:

http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/ [Broken]

I solved it by guessing a polynomial solution and then verifying the co-efficients, but in their solutions they have a more analytical approach where they let x = a/x. Now I'm a bit uncomfortable with this substitution so i let y = a/x, and preceeded to compute their solution. The point i get stuck is after they calculate the second derivative, they somehow eliminate all the a's. Could something explain that step? Its probaly trivial and I'm just not putnam material!

Last edited by a moderator: May 2, 2017
2. Nov 10, 2006

mattmns

They are just using what f '(x) is equal to, and a little bit of algebra.

3. Nov 11, 2006

Hurkyl

Staff Emeritus
Think of a similar situation: if you were told that

f(x) = x^2 + 3x + 5,​

would you be comfortable answering the question

What is f(1/x)?​