## Task 1 - 180811

The FDJ members Arnim, Bertram, Christian, Dieter, Ernst, and Fritz participated in a 400-meter race. No two of them finished at the same time.

Before the race, the following three predictions were made about the results (each participant is represented by the first letter of their first name):

| Place | 1. | 2. | 3. | 4. | 5. | 6. |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| 1st Prediction | A | B | C | D | E | F |
| 2nd Prediction | A | C | B | F | E | D |
| 3rd Prediction | C | E | F | A | D | B |

After the race, it turned out that in the first prediction, the places of exactly three runners were correctly given. No two of these three places were adjacent to each other. In the second prediction, no runner's place was correctly given. In the third prediction, the place of one runner was correctly given.

Give all possible sequences of the places achieved by the runners under these conditions!
I. If a ranking order meets the conditions, then for it: A did not take the 1st place and E did not take the 5th place, because in the 2nd prediction, the achieved place was not correctly given for any runner.

Furthermore, it follows: Since in the 1st prediction, the places were correctly given for exactly three runners and no two of these places were adjacent, B took the 2nd place, D the 4th place, and F the 6th place. Therefore, E and A did not take the 2nd, 4th, or 6th place. Thus, for all the places specified in the 3rd prediction except the first, the runners are incorrectly given.

Hence, C took the 1st place, as exactly one place in the 3rd prediction was correctly given. Since E consequently did not take the 1st place, only the 3rd place remains for E, and subsequently, the 5th place for A. Therefore, only the order C, B, E, D, A (*) can meet all the conditions of the problem.

II. This order indeed meets these conditions; because in the 1st prediction, the achieved places are correctly given for B, D, and F; these are the 2nd, 4th, and 6th places, and none of them are adjacent.

In the 2nd prediction, the achieved place is not correctly given for any runner; in the 3rd prediction, the runner (in this case, C) who achieved the 1st place is correctly given for one place, namely the 1st place.

Thus, the ranking order (*) meets all the conditions of the problem.