We are looking for a five-digit number with the following properties: it is a palindrome (i.e., it reads the same backward as forward), it is divisible by twelve, and in its representation, the digit 2 immediately follows the digit 4. Determine all possible numbers that meet the given conditions.

(M. Mach)

Hint. Determine how the digits 2 and 4 can be placed; for each case separately, then discuss the remaining conditions.
Five-digit palindromes in which the digit 2 appears immediately after the digit 4 are precisely the following:

$$
42 * 24, \quad * 424 *, \quad * 242 *, \quad 24 * 42 .
$$

For these cases, it is now sufficient to discuss divisibility by twelve. A number is divisible by twelve if and only if it is divisible by three and four, i.e., precisely when its digit sum is divisible by three and the last two digits are divisible by four.

The number 24 is divisible by four, so palindromes of the type $42 * 24$ are always divisible by four, and we only need to consider divisibility by three. The known digits have a digit sum of 12, which is divisible by three, so the asterisk in the middle must represent a multiple of three - $0, 3, 6$ or 9.

Palindromes of the type $* 424 *$ are divisible by four if and only if the last digit is 0, 4, or 8. Since it is a palindrome, the same digit will also be at the beginning, so the variant with zero is not suitable. After filling in the fours, the digit sum is 18, and after filling in the eights, it is 26. Therefore, only the palindrome 44244 is divisible by three.

Palindromes of the type $* 242 *$ are divisible by four if and only if the last digit is 0, 4, or 8. As in the previous case, we exclude zero and determine the digit sums: for fours, it is 16, and for eights, it is 24. Only the palindrome 82428 is divisible by three.

Since the number 42 is not divisible by four, palindromes of the type $24 * 42$ cannot be divisible by four, and thus not by twelve. The numbers that satisfy the given conditions are precisely the following:

42024, 42324, 42624, 42924, 44244, 82428.