How Do You Solve Putnam 2005 B3 with a Polynomial Guess?

[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/

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 probably trivial and I'm just not putnam material!

 

[mattmns]

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

 

[Hurkyl]

Now I'm a bit uncomfortable with this substitution

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)?​