  This paper studies Zeilberger's two prized constant term identities. For one
of the identities, Zeilberger asked for a simple proof that may give rise to a
simple proof of Andrews theorem for the number of totally symmetric self
complementary plane partitions. We obtain an identity reducing a constant term
in $2k$ variables to a constant term in $k$ variables. As applications,
Zeilberger's constant terms are converted to single determinants. The result
extends for two classes of matrices, the sum of all of whose full rank minors
is converted to a single determinant. One of the prized constant term problems
is solved, and we give a seemingly new approach to Macdonald's constant term
for root system of type BC.
