  In an influential 1877 paper, Zolotarev asked and answered four questions
about polynomial and rational approximation. We ask and answer two questions:
what are the best rational approximants $r$ and $s$ to $\sqrt{z}$ and
$\mbox{sign}(z)$ on the unit circle (excluding certain arcs near the
discontinuities), with the property that $|r(z)|=|s(z)|=1$ for $|z|=1$? We show
that the solutions to these problems are related to Zolotarev's third and
fourth problems in a nontrivial manner.
