  In the subject of optimal stopping, the classical secretary problem is
concerned with optimally selecting the best of $n$ candidates when their
relative ranks are observed sequentially. This problem has been extended to
optimally selecting the $k$-th best candidate for $k\ge 2$. While the optimal
stopping rule for $k=1,2$ (and all $n\ge 2$) is known to be of threshold type
(involving one threshold), we solve the case $k=3$ (and all $n\ge 3$) by
deriving an explicit optimal stopping rule that involves two thresholds. We
also prove several inequalities for $p(k,n)$, the maximum probability of
selecting the $k$-th best of $n$ candidates. It is shown that (i)
$p(1,n)=p(n,n)>p(k,n)$ for $1<k<n$, (ii) $p(k,n)\ge p(k,n+1)$, (iii) $p(k,n)\ge
p(k+1,n+1)$, and (iv) $p(k,\infty):=\lim_{n\to \infty} p(k,n)$ is decreasing in
$k$.
