3. Two two-digit numbers are written on the board. Andrey multiplied them, getting a four-digit number with the first digit 2. Pasha added them and got a three-digit number. If the first digit is erased from Andrey's number, Pasha's number is obtained. What numbers were written?

(l. S. Korechkova)
Solution. If the numbers $x$ and $y$ were recorded, then by the condition $xy = 2000 + x + y$, which means $xy - x - y + 1 = 2001$, or $(x-1)(y-1) = 2001$. Since $2001 = 3 \cdot 23 \cdot 29$ and $x, y$ are two-digit numbers, then $x-1$ and $y-1$ can be $23, 29, 23 \cdot 3$ or $29 \cdot 3$. Therefore, $x, y \in \{24, 30, 70, 88\}$. The pairs that satisfy the condition are $\{24, 88\}$ and $\{30, 70\}$.

Criteria. If one of the answers is incorrect - 0 points.

Finding one of the solutions - 1 point, finding both solutions - 2 points without a correct proof of the absence of other solutions.

If the solution process is correct but one of the solutions is lost due to a logical error in one of the cases of enumeration - 5 points.