## Task 1 - 120731

At a secondary school with exactly 500 students, there are mathematical-scientific, artistic, and sports working groups. The following is known about the participation of students in these working groups:

(1) Exactly 250 students are members of at least one sports working group.

(2) Exactly 125 students belong to at least one artistic working group.

(3) Exactly 225 students participate in at least one mathematical-scientific working group.

(4) Exactly 25 students attend at least one artistic and one sports working group.

(5) Exactly 75 students are members of at least one mathematical-scientific and one sports working group.

(6) Exactly 25 students participate in at least one mathematical-scientific and one artistic working group.

(7) Exactly 5 students belong to all three types of working groups.

Determine the number of all students at this school who

a) participate in exactly one type of these working groups,

b) do not participate in any of these working groups!
With the letters $M, S, K$ and the letter groups $M S, M K, S K, M S K$ let the numbers of students who participate in the respective working groups ( $M$ : math and science, $S$ : sports group, $K$ : artistic group) but not in the others be denoted. Let $N$ be the number of students who do not participate in any of the working groups. Then it follows:

(0) $N+M+S+K+M S+M K+S K+M S K=500$,

(1) $S+M S+S K+M S K=250$,

(2) $M K+S K+M S K=125$,

(3) $M+M S+M K+M S K=225$,

(4) $S K+M S K=25$,

(5) $M S+M S K=75$,

(6) $M K+M S K=25$,

(7) $M S K=5$.

Due to (7) and (4) or (5) or (6), we have
(8) $S K=20$
(9) $M S=70$
(10) $M K=20$

Due to (7) and (1), (8), (9) or (2), (8), (10) or (3), (9), (10), we have
(11) $S=155$
(12) $K=80$
(13) $M=130$

Due to (0), (7), (8), (9), (10), (11), (12), (13), $N=20$.

The number sought in a) is $S+K+M$; due to (11), (12), (13), it is 365. The number sought in b) is $N=20$.