Task 2 - 251012
Three mathematics teachers, each of whom was younger than 50 years old but older than 20 years old at the time, met at a joint birthday party.

Each of them had two children; remarkably, all six of these children had their birthdays on the same day.

During a conversation, the oldest of them said: "I am \(5 \frac{1}{2}\) times as old as my son and \(11\) times as old as my daughter. When my daughter is as old as my son is now, I will be \(6\) times as old as she is and \(4\) times as old as my son."

After a short consideration, the second mathematics teacher noted that these statements also applied to him and his older and younger child. Now the youngest of them also calculated and said: "It is strange, the same statements apply to me and my two children, even though we three teachers are of different ages."

Determine whether there are age specifications for the three teachers and their children that satisfy all these statements and whether the statements uniquely determine the age specifications! If this is the case, give the ages of the three teachers and their children!
\end{abstract}
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I. If the statement made by one of the three teachers is true and \(x\) is the age of his younger child, then \(11 x\) is the age of this teacher and thus, he is \(\frac{11}{2}\) times as old as his older child, with \(2 x\) being the age of the older child.

Therefore, the age statements for the three teachers can only be numbers divisible by 11. Since there are only three numbers divisible by 11 between 20 and 50, namely 22, 33, and 44, and the teachers are all of different ages, the statements can only be true if (*)

the oldest teacher is 44 years old and his children are 4 and 8 years old,

the second teacher is 33 years old and his children are 3 and 6 years old, the youngest teacher is 22 years old and his children are 2 and 4 years old.

II. If one of the teachers is \(11 x\) years old and his children are \(x\) and \(2 x\) years old, then he is \(5 \frac{1}{2}\) times as old as his older and 11 times as old as his younger child.

Furthermore, his younger child will be as old as his older one is now, namely \(2 x\) years, in exactly \(x\) years, and then his older child will be \(3 x\) years old and the teacher \(12 x\) years old, meaning he will be 6 times as old as his younger and 4 times as old as his older child.

With I. and II., it is proven that there are age statements for the three teachers and their children that satisfy all the statements mentioned in the problem and that these age statements are uniquely determined. They are as stated in \(\left({ }^{*}\right)\).