  Let $p$ be an odd prime. For nontrivial proper subsets $A,B$ of
$\mathbb{Z}_p$ of cardinality $s,t$, respectively, we count the number
$r(A,B,B)$ of additive triples, namely elements of the form $(a, b, a+b)$ in $A
\times B \times B$. For given $s,t$, what is the spectrum of possible values
for $r(A,B,B)$? In the special case $A=B$, the additive triple is called a
Schur triple. Various authors have given bounds on the number $r(A,A,A)$ of
Schur triples, and shown that the lower and upper bound can each be attained by
a set $A$ that is an interval of $s$ consecutive elements of $\mathbb{Z}_p$.
However, there are values of $p,s$ for which not every value between the lower
and upper bounds is attainable. We consider here the general case where $A,B$
can be distinct. We use Pollard's generalization of the Cauchy-Davenport
Theorem to derive bounds on the number $r(A,B,B)$ of additive triples. In
contrast to the case $A=B$, we show that every value of $r(A,B,B)$ from the
lower bound to the upper bound is attainable: each such value can be attained
when $B$ is an interval of $t$ consecutive elements of $\mathbb{Z}_p$.
