On the table lay six cards with digits $1,2,3,4,5$ and 6. Kamila made a three-digit number from three cards, which was greater than 500 and divisible by four. Filip made a three-digit number from the remaining three cards, which was divisible by both three and five. Kamila then added both three-digit numbers, and Filip noticed that this sum was a three-digit number that reads the same from left to right as from right to left.

Which numbers could Kamila and Filip have formed? Determine all possibilities. (L. Růžičková)
Filip's number must be divisible by five, so from the offered digits, it must use 5 in the units place. Therefore, 5 cannot appear in Kamila's number. At the same time, Kamila's number must be greater than 500, so it must have the digit 6 in the hundreds place.

Filip's number must be divisible by three, meaning its digit sum must be divisible by three. Therefore, the sum of the first two digits must give a remainder of 1 when divided by three. From the offered digits, Filip could form the numbers

$$
135, \quad 315, \quad 345, \quad 435 .
$$

Kamila's number must be divisible by four, meaning its last two digits must be divisible by four. From the offered digits, Kamila could form the numbers

$$
612, \quad 624, \quad 632 \text {. }
$$

At the same time, the digits used in Filip's number cannot appear in Kamila's number, and vice versa. Therefore, Filip and Kamila could form the following pairs of numbers:

| Filip | 135 | 315 | 345 | 435 |
| :---: | :---: | :---: | :---: | ---: |
| Kamila | 624 | 624 | 612 | 612 |
| Sum | 759 | 939 | 957 | 1047 |

The only sum that reads the same from left to right as from right to left is 939. Filip formed the number 315, and Kamila formed the number 624.

Evaluation. 1 point for determining that Filip's number ends with the digit 5 and Kamila's number starts with the digit 6; 2 points each for determining the possible numbers for Filip and Kamila that meet the divisibility conditions; 1 point for selecting the possible pairs, determining their sums, and solving the problem.

If the solver overlooks one of the conditions and lists another solution in addition to the only possible one (e.g., $632+145=777$), award no more than 4 points. For a correct solution without justification, award no more than 2 points.

Note. The sum of all six given digits is divisible by three, and the digit sum of Filip's number must be divisible by three, so the digit sum of Kamila's number must also be divisible by three. This observation in the above solution excludes the possibility of 632 among Kamila's numbers.