[Decimal numeral system]

A two-digit number, when added to the number written with the same digits but in reverse order, results in a perfect square. Find all such numbers.

#
Answer: $29,38,47,56,65,74,83,92$. Let $10 a+b$ be the desired number. According to the condition, the number $(10 a+b)+(10 b+a)=$ $11(a+b)$ is a square of some number $k$. Then $k$ is divisible by 11, which means $a+b$ is also divisible by 11. But $a+$ $b \leq 18$, so $a+b=11$.