Lenka is thinking of two two-digit numbers. One has both digits even, and the other has both digits odd. When she adds both numbers, she gets another two-digit number with the first digit even and the second digit odd. Additionally, Lenka revealed that all two-digit numbers are multiples of three, and one of the three odd digits is 9. What numbers could Lenka be thinking of? Find all possibilities.

(V. Hucíková)
The sum of an even and an odd number is always an odd number. However, in the final sum, the digit in the tens place is even, which is only possible if the sum of the digits in the units place is greater than 10. At the same time, we realize that for the two addends, this sum is at most 18.

Now we will determine which of the odd numbers can be 9:

- If it were the second digit in the result, then the sum of the digits in the units place would be 19, which is too much.
- If it were the first digit in one of the addends, then this addend would be at least 91. The addend with even digits is at least 20 (the tens place cannot be 0), so the resulting sum would not be a two-digit number.

The only remaining possibility is that 9 is the second digit in the addend with odd digits. We don't see any problem here yet, so we continue to investigate:

The addend with odd digits can be

19, 39, 59, 79, or 99.

Of these numbers, only 39 and 99 are multiples of 3. However, 99 is too large (adding any number would result in a three-digit number), so Lenka's number with odd digits can only be 39.

From this, we see that the second addend must not be greater than 60 (to keep the sum a two-digit number). This number must have only even digits, and from the first paragraph, we know that the units place must be at least 2. Therefore, the addend with even digits can be

22, 24, 26, 28, 42, 44, 46, or 48.

Of these numbers, only 24, 42, and 48 are multiples of 3. Thus, we get the following three possibilities:

$$
24+39=63, 42+39=81, 48+39=87
$$

In all cases, all conditions in the problem are met, so Lenka could have thought of any of the given pairs of addends.

Evaluation. 1 point for observing that the sum of the digits in the units place is greater than 10; 2 points for determining the number 39, including justification; 3 points for finding the addends 24, 42, and 48, including justification.