I am teaching a summer course and a student asked me a question he found online. It asks
"Show that there is not a continuous mapping $f:S^n\longrightarrow S^1$ with $f(-x)=-f(x)$ for all $x$."
The easy answer is that this would contradict the Borsuk-Ulam Theorem if we view $S^1$ as embedded in $\mathbb{R}^n$. Is there perhaps another way to prove this using degrees of maps with $S^1\subseteq S^n$? Or, can we pass to a map from $\mathbb{R}P^n$ to itself? Or is there at least some solution that doesn't require a big theorem?