Prove for every function that from sphere to real number has points $x$, $-x$ such that f$(x)=f(-x)$ I have not taken topology course yet. This is just the question that my undergrad research professor left us to think about. She hinted that I could use a theorem from Calculus.
So I reviewed all theorems in Calculus, and I found Intermediate Value Theorem might be helpful(?)... since it has some generalizations on topology. 
But I still don't know how to get started. If you could give me some hints or similar examples, that would be really helpful.
Thanks!

You may search borsuk–ulam theorem and get some details.
