## Task 1 - 120831

The FDJ group of class $8 a$ of a secondary school organized a sports competition. Before the competition began, each participant was asked to predict which three participants would achieve the best overall results and in what order.

When the prediction slips were evaluated, it turned out that only Annekatrin, Bernd, and Claudia were expected to take the top three places. The order Bernd - Annekatrin - Claudia was predicted exactly five times. In addition, the orders Bernd - Claudia - Annekatrin and Claudia - Annekatrin - Bernd were also expected, with one of these predictions being made exactly four times and the other exactly three times. No other order was predicted.

Points were awarded for the predictions, with one point given for each correctly predicted position. A maximum of three points could be achieved with a single prediction slip. The total sum of all points awarded was 17. Bernd won the competition, contrary to most predictions, but the top three places were indeed taken by Annekatrin, Bernd, and Claudia.

Who won the competition? Who took second and who took third place? How often was the prediction Bernd - Claudia - Annekatrin given in total?
In the following, let the names of the three participants noted on the betting slips be abbreviated with A, B, and C, and the ranking be indicated by the order of these three letters.

Since A, B, and C took the first three places and B did not come in first, only the following rankings are possible: (1) ABC, (2) ACB, (3) CAB, (4) CBA.

Now, a table can show how many points could have been awarded in cases (1) to (4) for each of the possible tip distributions according to the problem statement:

1. Possibility of tip distribution

| Ranking | ABC | ACB | CAB | CBA |
| :--- | :--- | :--- | :--- | :--- |
| Tip BAC 5 times | $0+0+5$ | $0+0+0$ | $0+5+0$ | $0+0+0$ |
| Tip BCA 4 times | $0+0+0$ | $0+4+0$ | $0+0+0$ | $0+0+4$ |
| Tip CAB 3 times | $0+0+0$ | $0+0+3$ | $3+3+3$ | $3+0+0$ |
| Sum | 5 | 7 | 14 | 7 |

2. Possibility of tip distribution

| Ranking | ABC | ACB | CAB | CBA |
| :--- | :--- | :--- | :--- | :--- |
| Tip BAC 5 times | $0+0+5$ | $0+0+0$ | $0+5+0$ | $0+0+0$ |
| Tip BCA 3 times | $0+0+0$ | $0+3+0$ | $0+0+0$ | $0+0+3$ |
| Tip CAB 4 times | $0+0+0$ | $0+0+4$ | $4+4+4$ | $4+0+0$ |
| Sum | 5 | 7 | 17 | 7 |

As the table shows, the total point score of 17 is achieved only if Claudia won the competition, Annekatrin came in second, and Bernd came in third, and the tip BCA was given exactly 3 times.