Question 24
What is the number of ordered triples $(a, b, c)$ of positive integers, with $a \le b \le c \le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a, b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6