A sports club has 80 members and three sections: football, swimming, and chess. We know that

a) there is no athlete who plays chess but does not swim;

b) the football players who do not swim all play chess;

c) the combined number of members in the chess and football sections is the same as the number of members in the swimming section;

d) every member is part of at least two sections.

How many members are in all three sections at the same time?
If we add up the number of members of the chess and football sections, according to condition d) we have considered all members of the club, and even those who are members of all three sections, twice. However, according to c), with this method we only obtained the number of members of the swimming section; consequently, no one should have been counted twice, meaning no one can be a member of all three sections at the same time.

To solve the problem, it is also necessary to show that the club can exist, that is, the conditions are not contradictory. For example, let each of the 80 members be a swimmer, of whom 40 are chess players, and the other 40 are members of the football section. The fulfillment of a), b), e), and d) can be easily verified.

Notes: 1. It also follows from the above that the club cannot have any non-swimming members.

2. We did not need the conditions a) and b) for the solution.