= Solution
Define $g(p)=f(p)-f(-p)$ on the <sphere>. It is <continuous> and satisfies $g(-p)=-g(p)$. If $g(p)=0$ at a chosen point, we are done. Otherwise choose a <unit vector> $q$ perpendicular to $p$ and follow the half <great circle>
$$
\gamma(t)=\cos(\pi t)p+\sin(\pi t)q,\qquad0\leq t\leq1.
$$
The <Pythagorean theorem> shows that $\gamma(t)$ lies on the <sphere>, and its endpoints are $p,-p$. The <continuous function> $g\circ\gamma$ has opposite signs at the endpoints. The <Intermediate value theorem> therefore gives a zero $t_*$, so
$$
\boxed{f(\gamma(t_*))=f(-\gamma(t_*))}.
$$
This is a scalar equal-value assertion for an <antipodal pair>; it uses only a <path> and the <Intermediate value theorem>.
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