The anatomy of a Putnam problem

In this talk, I shall bring into perspective an analysis problem given at the Putnam Mathematical Competition in 2007. The question B2 on the exam reads as the following: 

               Suppose that $f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $\int_0^1f(x)dx=0$. 

 

 

Following Polya's approach on how to solve a problem (https://math.berkeley.edu/~gmelvin/polya.pdf), we "dissect" the above question by giving several solutions and by relating it to other similar problems or problems with similar ideas.  In fact, we go back to one of Polya's original problems and show how ideas can be applied in our situation. Last but not least, we also discuss some variations of our main question.