(i) State, but do not prove, the Lefschetz fixed point theorem.

(ii) Show that if $n$ is even, then for every map $f: S^{n} \rightarrow S^{n}$ there is a point $x \in S^{n}$ such that $f(x)=\pm x$. Is this true if $n$ is odd? [Standard results on the homology groups for the $n$-sphere may be assumed without proof, provided they are stated clearly.]