## Task 3 - 120723

The ages of four tennis players, Arnold, Bruno, Christoph, and Detlef, are known as follows:

(1) All four players together are 100 years old.

(2) Arnold and Bruno together are exactly as old as Christoph and Detlef together.

(3) Christoph is older than Detlef.

(4) If all possible "doubles" (groups of two players) that can be formed from the four players are considered, exactly one of these "doubles" consists of two players of the same age.

(5) The oldest of the four players is four years older than the youngest.

How old is each of the four players? (All ages are given in full years)
The ages of the four players, given in years, are denoted in sequence by $a, b, c, d$. According to the problem:
(1) $a+b+c+d=100$
(2) $a+b=c+d$
(3) $c>d$

From (1) and (2), it follows that (6) $a+b=c+d=50$.

If $a=c$ or $a=d$ or $b=c$ or $b=d$, then from (2) it would follow that $b=d$ or $b=c$ or $a=d$ or $a=c$, which contradicts (4). Therefore, and due to (3), it follows from (4) and (6) that $a=b=25$ must hold.

Furthermore, due to (6) and (3), $c>50-c$, so $c>25$, and therefore $d=50-c<25$.

Thus, Christoph is the oldest and Detlef is the youngest of the four players. Due to (5), it follows that $c-d=4$, and together with (6), this results in $c=27$ and $d=23$.

Therefore, Christoph is 27 years old, Arnold and Bruno are each 25 years old, and Detlef is 23 years old.