Putnam problem of the day (by the HMD dated March 1, 2008)

pdf: included in Caderno (see “caderno” page) 

On March 1st, 2008, the Putnam problem of the day displayed on the  Harvard’s Math Department site was stated as follows:

“ Evaluate

Express your answer in the form

,

where are integers. “ 

Solution

To evaluate the radicand I start by seeing that the continued fraction

satisfies

.

Thus,  since  the only solution left is

.

A few algebraic manipulations give

hence

In order to have

or equivalently,

with integers,  should also be an integer; therefore should be even. I assume that On the other hand  should be  Thus,

Since, for 

,

this possibility is excluded. It remains 

Now I confirm

So, the solution I came was

Remark: The calculation of

can be done by hand as follows

Update March, 20: you can compare with this solution   (Putnam 1995, Problem B5 )

Addendum of March 7, 2009: Comment/Proof of the convergence of the continued fraction by Vishal Lama (comment dated March 7, 2009)

“ In the solution presented in your post, denotes an expression that we can’t assume, beforehand, is a finite number. may perhaps be infinite! Therefore, the way to go about computing the expression (the infinite continued fraction) given in the problem is as follows.

The infinite continued fraction is defined as the limit of the sequence , where and for all . Then, we show that the sequence is bounded from below ( for all , which can be shown by a simple induction) and that it is also strictly decreasing (which can be shown using induction, again).  Now, we invoke the Monotone Convergence Theorem to conclude that the sequence does indeed have a (finite) limit, which we can now denote by . Once we establish that (which is the infinite continued fraction!) is finite, we can compute the way you did in your solution. Basically, we have to go through all that trouble just to prove that the given infinite continued fraction is indeed finite! Only after that can the computation begin! “

Part of my reply was: “I did not prove the convergence of the continued fraction. Thanks for doing it.
Basically I assumed that convergence based on a certain numerical evidence, but of course this evidence proves nothing.”

ADDENDUM of May 5, 2010: Alternatively we can prove the convergence applying the Śleszýnki-Pringsheim Theorem (a reference here,  wikipedia): if for all naturals , , then the continued fraction  converges. Since and , the theorem hypothesis is satisfied:

.

 

 

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