## Task 4 - 170824

Dieter tells his classmates:

"My brother Fritz is only half as old as I am. If you multiply the number of his years by itself, you get my father's age. My mother is three years younger than my father. All together, we are 87 years old."

Determine the age of all 4 people! (Only the completed years of life are to be considered.)
The father's age is a square number less than 45, and since the combined age of the father and mother does not amount to 87 years or more. As the father is the oldest of the four people, his age is at least a quarter of 87 years, so it is greater than 21. Between 21 and 45, the only square numbers are 25 and 36.

Case 1: Suppose the father's age is 25 years. Then Fritz would be 5 years old, Dieter 10 years old, and the mother 22 years old. The total age of all family members in this case would not be 87 years. Therefore, the father is not 25 years old.

Case 2: Suppose the father's age is 36 years. Then Fritz is 6 years old, Dieter 12 years old, and the mother 33 years old. Together, they are $36 + 6 + 12 + 33 = 87$ years old, as required.

Thus, these age assignments are the only ones that fit.

Solutions of the 2nd Round 1977 adopted from [5]

### 5.19.3 3rd Round 1977, Class 8