## Task 1 - 130731

The age information (in full years of life) of a family (father, mother, and two children) is as follows:

(1) The sum of all four ages is 124.

(2) The father and mother together are three times as old as their two children combined.

(3) The mother is more than twice as old as the oldest of the two children.

(4) The difference obtained by subtracting the mother's age from the father's age is nine times as large as the difference obtained by subtracting the younger child's age from the older child's age.

How old is each of the four family members?
Due to (1) and (2), the quadruple sum of the ages of the two children is 124; therefore, the children together are 31 years old, and the parents together are 93 years old.

Since the ages of the four individuals are given in whole years and since 31 is an odd number, one of the children's ages is even, and the other is odd. Therefore, the difference in the ages of the two children is an odd number.

If this difference were 3 years or more, then the difference in the parents' ages would be 27 years or more. In that case, one child would be 17 years or older, the other child 14 years or younger, the mother 33 years or younger, and the father 60 years or older.

Because $2 \cdot 17 > 33$ and (3), this possibility is ruled out. Therefore, the difference in the children's ages is 1 year, and the difference in the parents' ages is 9 years. From this, it follows:

The father is 51 years old, the mother is 42 years old, the oldest child is 16 years old, and the other child is 15 years old.