\section*{Task 3 - 121233}

Three schools, one each from Adorf, Bedorf, and Cedorf, conducted a track and field competition at a county sports festival. In each discipline, each school provided exactly one participant. A reporter interviewed a spectator after the competition:

Reporter: "Who won the entire competition?"

Spectator: "Adorf won the long jump, but the entire competition was won by Bedorf with 22 points. Adorf and Cedorf each scored 9 points."

Reporter: "How were the points distributed?"

Spectator: "In each discipline, the first place received a certain number of points, the second place received fewer points, and the third place received even fewer points, but at least one point. This distribution was the same for all disciplines. All point values were integers."

Reporter: "In how many disciplines did the competition take place?"

Spectator: "I don't know."

Reporter: "Who won the shot put?"

Spectator: "I don't know, but shot put was included.

Determine whether the following two questions can be definitively answered based on these (all assumed to be true) statements, and provide all answers that are consistent with these statements!

a) Which of the three schools won the shot put?

b) Which school took second place in the long jump?

(It is known that in each discipline, a clear ranking of the competition participants was determined.)
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Let \(k\) denote the number of competitions and \(P\) the total number of points to be distributed for each competition. Since the total number of points for all competitions combined is 40 (as given by \(9+22+9\)), we know that \(k \cdot P = 40\) must hold. Furthermore, \(P \geq 6 (3+2+1)\), because the point values for the individual placements are different positive integers, as stated. This restricts \(k\) to the three cases \(k=2, 4, 5\), which we will examine below.

1. Case: \(k=2, P=20\)

In this case, ADorf would have to receive more than 10 points for winning the long jump alone, although his total score is only 9, which is a contradiction!

2. Case: \(k=4, P=10\)

For a single win in a competition, there must be at least 6 points, so that BeDorf can reach his total of 22 points. However, this is also the maximum value for such a win, otherwise the scores of 9 for ADorf and CeDorf would not be possible. This implies the point distribution 6, 3, 1 for the three places in each competition, but then BeDorf would not be able to reach his 22 points.

3. Case: \(k=5, P=8\)

For a single win in a competition, there must be at least 5 points, so that BeDorf can reach his total of 22 points. This is also the highest value for a win for which a decomposition of \(P\) into 3 different positive integers is possible, namely in this case \(P=8=5+2+1\). Apart from the order of the summands, there is then a compatible decomposition of the score for each of the three contestants, namely:

ADorf: \(5+1+1+1+1=9\)

BeDorf: \(2+5+5+5+5=22\)

CeDorf: \(1+2+2+2+2=9\)

In particular, the answers to the above questions are also fixed:

a) BeDorf won the shot put, because it won all competitions except the long jump.

b) For the same reason, BeDorf also placed second, which was in the long jump.