\section*{Task 1 - 201031}

In a district of Berlin, 50 students from the Olympic class 10 participated in the 2nd stage of the OJM. The following statements refer to this group of participants:

(1) An equal number of boys and girls participated.

(2) Exactly 24 of the participants, including exactly 15 boys, were members of a mathematics club.

(3) Exactly 13 of the participants received prizes or certificates of recognition. (These 13 participants will be referred to as "prize winners" in the following.)

(4) Exactly 12 of the prize winners were members of a mathematics club.

(5) Exactly 6 of the prize winners were girls.

(6) Only girls who were members of a mathematics club were prize winners.

Determine the number of participating boys who neither received a prize nor were members of a mathematics club!
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![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1432.jpg?height=380&width=368&top_left_y=938&top_left_x=276)

Boys

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1432.jpg?height=386&width=374&top_left_y=932&top_left_x=681)

Girls
From the problem statement, it immediately follows for the number of girls \(M\) and the number of boys \(J\): \(M=J=25\). For the prize winners, it is stated that 6 girls and 6 boys who are also AG members won prizes. Since no female AG member was left without a prize, a boy from an AG must not have won a prize.

Entering the data into two diagrams then results in: 15 boys are not prize winners and are not in any AG.