## Task B-1.8.

The sum of a two-digit number and the number with the same digits but in reverse order is a perfect square. Determine all such numbers!
## Solution.

Let the desired number be denoted by $\overline{x y}$. Then,

$$
\begin{aligned}
& \overline{x y} + \overline{y x} = n^{2} \\
& 10 x + y + 10 y + x = n^{2} \\
& 11 \cdot (x + y) = n^{2}
\end{aligned}
$$

The sum of the digits must be 11, which can be achieved in the following ways:

$$
2+9, 3+8, 4+7, 5+6, 6+5, 7+4, 8+3, 9+2
$$

The desired numbers are: $29, 38, 47, 56, 65, 74, 83$ and 92.

(1 point)

## SCHOOL/CITY COMPETITION IN MATHEMATICS

## 2nd grade - high school - B category

February 4, 2010.

IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH IN AN APPROPRIATE MANNER.