2. Jana and Žana wrote down their ages. They used the same digits, but in reverse order. In 5 years, Jana will be twice as old as Žana will be then. How much older is Jana than Žana?
I/2. Denote that Jana is $10 a+b$ years old, and Žana is $10 b+a$ years old. In 5 years, Jana will be twice as old as Žana, so we have $10 a+b+5=2 \cdot(10 b+a+5)$, from which we get $8 a=19 b+5$. Since the value on the left side is at most 72, $b$ can be 1, 2, or 3. However, only $b=1$ is valid, which gives us $a=3$. Jana is 31 years old, and Žana is 13. Jana is 18 years older than Žana.

Notation that Jana is $10 a+b$ years old, and Žana is $10 b+a$ years old ................... 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=51&width=1633&top_left_y=1568&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=49&width=1627&top_left_y=1620&top_left_x=223)

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=51&width=1633&top_left_y=1668&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=57&width=1633&top_left_y=1713&top_left_x=220)

Conclusion: Jana is 18 years older than Žana ...................................................................