Consider a continous map from $S^2$ to $C$.
Is it true that there exists 3 points equially spaced on a great circle, $x_1,x_2,x_3$, such that if $w$ is the third root of unity, $f(x_1)+wf(x_2)+w^2f(x_3)=0$?
More generally I'm asking this if we take nth unity roots.
Maybe I should add slight motivation: In my topology course we were shown a proof of borsuk ulam that goes through defining $g(x)=(f(x)-f(-x))/(|f(x)-f(-x)|)$, then by looking at it on the great circle, we can lift it to a function to $R$ satisfying (here we view $g$ as a function from $[0,1]$ instead of from the circle, and taking $x+1/2$ mod 1 in the next expression) $g(x+1/2)=n+1/2+g(x)$ for some natural $n$, but since it is continous, this is the same $n$ for all $x$. In particular $g(1)=2n+1+g(0)$ and thus this is a nontrivial path on the circle, but it homotopic to the constant one via returning to the sphere and wrapping the circle around to a constant function.
Then notice the argument after the lift works the same when we know $g(x+1/3)=n+1/3+g(x)$. However there is no direct analong we can do to reach the part of lifting, because that would involve choosing for each point a great circle it is contained in a way that partions the sphere into great circles which is clearly impossible.
If this turns out to be false, is there a space we can do this trick on to get this cute result?