## Task 2 - 110732

In a class of 28 students, all students participate in extracurricular sports, and each student participates in at least one of the following four sports: football, athletics, swimming, and gymnastics, with at least 1 student in each of these sports. No student participates in a sport not listed here.

It is known about the students in this class:

(1) Each student participates in at most two sports.

(2) Exactly 18 students participate in exactly one sport.

(3) Half of the students who do athletics also participate in gymnastics.

(4) Every swimmer participates in two sports, with the other sports being equally represented.

(5) The number of students who only do gymnastics is equal to the number of students who only play football.

(6) The set of students who both do gymnastics and play football is empty.

(7) The number of students who do both gymnastics and athletics is equal to the number of the remaining students who also participate in two sports.

Determine the number of students in this class who participate in

a) football b) athletics c) swimming d) gymnastics!
For the abbreviation, we denote the students who participate in only one sport with F, L, S, T, and the students who participate in two sports with FL, FS, FT, LS, LT, ST, according to the initial letters of the sports. Then it follows:

(8) Due to (2), exactly 18 students participate in exactly one sport.

(9) Due to (1) and (8), and because the class has 28 students, exactly 10 students participate in exactly two sports.

(10) Due to (9) and (7), there are exactly 5 LT.

(11) Due to (9), (10), (4), and because there is at least 1 swimmer according to the problem statement, there are exactly 1 FS, 1 LS, 1 ST. If there were 2 of each, the total would exceed 10 because $5+6=11>10$.

(12) Due to (9), (10), (11), and (6), there are exactly 2 FL.

(13) Due to (10) and (3), there are exactly 10 track and field athletes, so due to (10), (11), and (12), there are exactly 2 L.

(14) Due to (8), (13), (4), and (5), there are exactly 8 P and 8 T. Therefore, in this class, there are exactly 11 footballers (namely 8 F, 2 FL, 1 FS), 10 track and field athletes (namely 2 L, 2 FL, 1 LS, 5 LT), 3 swimmers (namely 1 FS, 1 LS, 1 ST), and 14 gymnasts (namely 8 T, 5 LT, 1 ST).