Insights of Approaching certain Mathematical Problems

1. I would also like to add that changing the stuff such that the things we are interested in don’t change is really what makes maths a dynamic subject.

2. Looking that the $f_n$ and $g_n$ also make wonder about the following: we want to prove a certain property about $f_n$ and $g_n$ is like the canonical function for this holds and the transformation provides a path between f to the canonical member of the class of polynomials with the property we are proving. I believe that this is a general phenomenon. Stated more clearly, say $F$ is the family of objects which have a certain property and say $f$ is the canonical representative of this family $F$. Then I have noticed in many places that there are canonical paths to connect all the members of $F$ to $f$. These nice canonical paths in turn allow us to bring objects into $F$ and hence help define $F$ more explicitly. Faith in this phenomenon also allow us to come up with reasonable conjectures about existence of such paths.
Conincidentially, I had personal experience on these lines recently, which is also elementary:
Real Polynomials with positive coefficients have no positive roots. These are the canonical objects in the family of all polynomials with no positive roots. Now, the phenomenon about canonical pathsi mentioned above, made me guess the following:
Any poly. with no positive roots can be multipiled with another polynomial such that the result only has positive coefficients. It actually turns out that this was true and is a good elementary exercise.

3. on a sidenote, another approach to solve the problem which might be more natural to ppl who are interested in the trick rather than the understanding of the phenomenon can be to just try to treat $f_n$ as a summation task. This road will lead one to sum the polynmial up by any of the various simple summation techniques and actually yield the same $g_n$


I have been solving Olympiad problems for sometime and I want to share my insight on how to attack certain problems. I will of course share an example for better understanding and quality of the information.I would also like to add one more point, one problem can be solved in numerous ways using various tools and techniques.In mathematics this is such an wonderful phenomena. The technique I am presenting here has helped me to proceed towards the solution without involving much complicacy on the way and also gain some insight which I also be sharing.Actually this is the whole point of the blog to share the insight and it is nothing new but I gained it on my own through problem solving.

Let us take a problem first and the method by which I have been able to solve it and it is of course not that difficult but it…

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