= Solution
Using the supplied counting function, an appropriate <first-order sentence> is
$$
\boxed{\forall x\Bigl(\bigl(\mathit{FacultyMember}(x)\land\mathit{GreaterOrEqual}(\mathit{CountSupervise}(x),2)\land\exists y\,\mathit{Teach}(x,y)\bigr)\to\mathit{Busy}(x)\Bigr).}
$$
Here the counting comparison expresses the threshold and the <existential quantification> expresses teaching at least one course. The word “if” supplies this direction of <logical implication>; it does not assert that these are the only reasons a faculty member can be busy. With the intended meanings of the nonlogical symbols, no further counting axioms are needed for this translation. If those meanings were unspecified, the displayed <first-order sentence> alone would not define arithmetic or connect the counting function to the supervision relation.
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