A number is said to be a palindrome if its reading from right to left is the same as from left to right. For example, the numbers 23432 and 18781 are palindromes. How many four-digit palindromic numbers are divisible by 9?
A four-digit palindrome number is of the form $a b b a$, where $a$ is a digit between 1 and 9 and $b$ is a digit between 0 and 9. Since the number is divisible by 9, the sum $2 a+2 b=2(a+b)$ of its digits is divisible by 9, that is, $a+b$ is divisible by 9. Since $1 \leq a+b \leq 18$, the only options are $a+b=9$ or 18. If $a+b=18$, necessarily $a=b=9$. If $a+b=9$, we have the following nine solutions.

$$
\begin{array}{lll}
a=1 \text { and } b=8 & a=2 \text { and } b=7 & a=3 \text { and } b=6 \\
a=4 \text { and } b=5 & a=5 \text { and } b=4 & a=6 \text { and } b=3 \\
a=7 \text { and } b=2 & a=8 \text { and } b=1 & a=9 \text { and } b=0
\end{array}
$$

Thus, there are ten four-digit palindrome numbers divisible by 9, namely, 1881, $2772,3663,4554,5445,6336,7227,8118,9009$ and 9999.