No homework, genuinely interesting problems, here are two of mine.

>Physics
Give a solid argument as to why atomic bonds can be approximated as springs between the two particles. (Hint: Van der waals force)

>Math
Let $f: A \subset R^{2} \rightarrow R$ be a continuous map, prove that the radius $r$ of the closed ball $B_{r}(0)$ which encloses $f(A)$ is such that $r=max \{ |min (f)| , |max(f)| \}$

>Chemistry
Give a solid excuse as to why you've messed up the titration so many times, jokes aside.

>Biology
The Polymerase Chain Reaction is a well known method to test the presence of viruses in a sample, could you in theory develop similar method to increase the replication rate of cancer cells?

Sorry if any seem uninteresting or trivial, post yours and see how they do.