jmc

Let $AB$ be a two-digit integer with the property that $AB$ is equal to the sum of its first digit plus its second digit plus the product of its two digits. Thus we have $10A+B=A+B+AB\iff 9A=AB$. Now since $AB$ is a two-digit integer, $A\ne 0$, so we can divide both sides by $A$ to get $9=B$. Therefore, 19, 29, 39, 49, 59, 69, 79, 89, and 99 all have the property. Their arithmetic mean is $\frac{\frac{9(19+99)}{2}}{9}=\frac{19+99}{2}=\boxed{59}$.