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

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In summary, the conversation discusses a Putnam competition question and the different approaches used to solve it. One person solved it by guessing a polynomial solution and verifying the coefficients, while the other used a substitution method. The second person was unsure about a step in the solution where all the variables were eliminated, but it was explained as using the value of the first derivative and algebra.
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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!
 
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They are just using what f '(x) is equal to, and a little bit of algebra.
 
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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)?​
 

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

What are Putnam questions for fun?

Putnam questions for fun are a series of mathematical and logical puzzles created by mathematician and philosopher Hilary Putnam. They are designed to challenge the problem-solving skills and critical thinking abilities of individuals.

How do you solve Putnam questions for fun?

There is no one set method for solving Putnam questions for fun as they can vary in difficulty and subject matter. However, some general tips for solving them include breaking down the problem into smaller parts, considering different approaches, and revisiting the problem if you get stuck.

Are Putnam questions for fun only for mathematicians?

No, Putnam questions for fun are not limited to mathematicians. They can be enjoyed and solved by anyone with an interest in problem-solving and logic. In fact, they were originally created for non-mathematicians to encourage critical thinking and creativity.

Can Putnam questions for fun be solved without any formal mathematical training?

Yes, some Putnam questions for fun can be solved without any formal mathematical training. However, having knowledge of basic mathematical concepts and principles can be helpful in approaching and solving some of the more challenging questions.

Are there any resources available for Putnam questions for fun?

Yes, there are various resources available for Putnam questions for fun, including books, online forums, and practice exams. These resources can provide helpful tips and techniques for solving the questions and also offer a variety of practice problems to improve problem-solving skills.

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