Task B-1.3. Determine the four-digit number which is 594 greater than the number obtained by swapping the two-digit beginning and the two-digit end (moving the first two digits to the end). The difference of the squares of the two-digit beginning and the two-digit end of the given number is 204.
## Solution.

Let the two-digit beginning be denoted by $x$, and the two-digit ending by $y$. Then we have

$$
\begin{aligned}
& x \cdot 100 + y = y \cdot 100 + x + 594 \\
& 99 x - 99 y = 594 \\
& x - y = 6 \\
& \text{and} \\
& x^2 - y^2 = 204 \\
& (x - y)(x + y) = 204 \\
& 6(x + y) = 204 \\
& x + y = 34
\end{aligned}
$$

Solving the system

$$
\left\{\begin{array}{l}
x - y = 6 \\
x + y = 34
\end{array}\right.
$$

we get $x = 20, y = 14$.

The desired number is 2014.

(2 points)