 Solution

For a sphere self-map, the Lefschetz number is $L(f)=1+(-1{)}^n\mathrm{deg}f$. At degree zero this is $1$, so the Lefschetz fixed-point theorem supplies $x$ with $f(x)=x$.

Let $A:S^n\to S^n$ be the antipodal map, $A(z)=-z$, of degree $(-1{)}^{n+1}$. The composite $Af$ also has degree zero, hence Lefschetz number one, so it has a fixed point $y$. The equation $A(f(y))=y$ is $f(y)=-y$. ConsequentlyThe two points are distinct, since a unit vector is never its own negative.