RU-ALLRUSSIAN 1976 Problem 15

S is a sphere with unit radius. P is a plane through the center. For any point x on the sphere $\mathrm{f}(\mathbf{x})$ is the perpendicular distance from $\mathbf{x}$ to P. Show that if $\mathbf{x}$ , $\mathbf{y}$ , $\mathbf{z}$ are the ends of three mutually perpendicular radii, then $\mathrm{f}(\mathbf{x})^2 + \mathrm{f}(\mathbf{y})^2 + \mathrm{f}(\mathbf{z})^2 = 1$ (\). Now let $\mathrm{g}(\mathbf{x})$ be any function on the points of S taking non- negative real values and satisfying (\). Regard the intersection of P and S as the equator, the poles as the points with $\mathrm{f}(\mathbf{x}) = 1$ and lines of longitude as semicircles through both poles. (1) If $\mathbf{x}$ and $\mathbf{y}$ have the same longitude and both lie on the same side of the equator with $\mathbf{x}$ closer to the pole, show that $\mathrm{g}(\mathbf{x}) > \mathrm{g}(\mathbf{y})$ . (2) Show that for any $\mathbf{x}$ , $\mathbf{y}$ on the same side of the equator with $\mathbf{x}$ closer to the pole than $\mathbf{y}$ we have $\mathrm{g}(\mathbf{x}) > \mathrm{g}(\mathbf{y})$ . (3) Show that if $\mathbf{x}$ and $\mathbf{y}$ are the same distance from the pole then $\mathrm{g}(\mathbf{x}) = \mathrm{g}(\mathbf{y})$ . (4) Show that $\mathrm{g}(\mathbf{x}) = \mathrm{f}(\mathbf{x})$ for all $\mathbf{x}$ .