Suppose $f: S^1 \to S^1$ is a continuous function which is homotopic to a constant function $g: S^1 \to S^1$, $g(x) = z$ for all $x \in S^1$. Prove there exists $x\in S^1$ such that the arclength between $x,f(x)$ is $\pi/6$.
So far, I've thought about using the covering map $p(t) = e^{2\pi it}$. The compositions $\gamma = f \circ p$ and $\delta = g \circ p$ are paths in $S^1$. $\delta$ is a constant path in $S^1$, and $\gamma$ is a closed path since $\gamma(0)=f(p(0))=f(1)=f(p(1))=\gamma(1)$ (we identify $(1,0)$ on $S^1$ with $1+0i$ on the unit circle in the complex plane).
Well, I'm not really sure how to proceed from here (it's not really much to begin with). Would appreciate any help.