jmc

We break the belt into six pieces, three where the belt touches no circle and three where it does.

First consider the portion of the belt that does not touch a circle. Each segment is the length of two radii, or $20$ cm. There are three such segments, or $60$ cm in total.

Now consider the portion of the belt that does touch a circle. Because there are three circles, the belt will touch each circle for $\frac{1}{3}$ of its circumference. Since it does this three times, this is the length of these segments combined, which is the circumference of a full circle, which is $20\pi$ cm for a circle of radius $10$ cm.

Therefore the length of the belt is $60+20\pi$ cm. From this we conclude that $a=60$ and $b=20,$ and so $a+b=\boxed{80}.$