## Task 34/82

Mr. X shares the following about the ages of his relatives:

1. My mother is twice as old as I am.
2. My father is older than my mother by the cross sum of my age.
3. The age of my youngest aunt is obtained by adding my mother's age to my age with the units and tens digits swapped.
4. My oldest aunt is older than the youngest aunt by the cross sum of my age.
5. My uncle is one year older than my oldest aunt and is celebrating a "round" birthday.
6. All my relatives are younger than 100 years old.

How old are the six people?
Mr. X is $10 a+b$ years old $(a ; b \in N, 0 \leq a, b \leq 9)$. Then the age is

1. of his mother $20 a+b$
2. of his father $20 a+2 b+a+b=21 a+3 b$
3. of his youngest aunt $20 a+2 b+10 b+a=21 a+12 b$
4. of his oldest aunt $21 a+12 b+a+b=22 a+13 b$
5. of his uncle $22 a+13+b+1=10 k$ with $k \in N, k \geq 2$ (since surely $a$ and $b$ are not both 0), $k10$ )
3. $a=2, b=5$ : (excluded because $k=10$ )
4. $a=3, b=1$ : (is a solution!)
5. $a=4, b=7$; (excluded because $k>10$ ).

Mr. X is therefore 31 years old, his mother 62, his father 66, his youngest aunt 75, the oldest aunt 79, and the uncle is celebrating his 80th birthday.