13. 3 (CMO 2000) Let $a, b, c$ be the three sides of $\triangle ABC$, $a \leqslant b \leqslant c$, and $R$ and $r$ be the circumradius and inradius of $\triangle ABC$, respectively. Let $f=a+b-2R-2r$, try to determine the sign of $f$ using the size of angle $C$.


13. $3 a=2 R \sin A, b=2 R \sin B, r=4 R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$. Substituting these, we can simplify to
$$
\begin{array}{l}
f=2 R\left(\sin A+\sin B-1-4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}\right)=\cdots= \\
2 R\left(\cos \frac{C}{2}-\sin \frac{C}{2}\right)\left(2 \cos \frac{B-A}{2}-\cos \frac{C}{2}-\sin \frac{C}{2}\right) .
\end{array}
$$

Let $A \leqslant B \leqslant C$, and by discussing the signs, we get: $f>0 \Leftrightarrow C<\pi / 2$.