Baltic Way 1995 · Problem 4

Solution:

Let John's age be $10a + b$ where $0 \leq a, b \leq 9$. Then Mary's age is $10b + a$, and hence $a > b$. Now

$$(10a + b)^2 - (10b + a)^2 = 9 \cdot 11 (a + b)(a - b).$$

Since this is the square of an integer, $a + b$ or $a - b$ must be divisible by $11$. The only possibility is clearly $a + b = 11$. Hence $a - b$ must be a square. A case study yields the only possibility $a = 6, b = 5$. Thus John is $65$ and Mary $56$ years old.