AMC 10 A

Solution:

The value of a two-digit cuddly number $\underline{ab}$ is $10a + b$. Therefore $10a + b = a + b^2$, so $9a = b(b - 1)$. It follows that $9$ divides $b(b - 1)$. If $3 \mid b$ and $3 \mid (b - 1)$, then $3 \mid 1$, which is a contradiction. Thus either $9 \mid b$ or $9 \mid (b - 1)$. Furthermore, $a \neq 0$ implies $b > 1$. Therefore the only possible value for $b$ is $9$. Then $9a = 9 \cdot 8$, so $a = 8$, and the only two-digit cuddly number is $89$.