[borisp]

I guess some people who are familiar with theorems don't like to find their most elegant solutions. There are elementary arguments that verify Brouwer's fixed point theorem, although every one I know requires smoothness of f. I figured it might be interesting to pose it, in an effort to tease out one of those arguments from someone who hasn't had exposure to heavy machinery, and who wasn't aware that this is a standard theorem of algebraic topology.

And yes, the first problem is easy (from an old Putnam exam I think?) I use it to illustrate the concept of "proof" to my students on the first day of lecture in a proof based course.