Adéla wrote two positive integers on the board and gave Lukáš and Petr the task of determining the positive difference of the squares of these two numbers. Instead, Lukáš determined the square of the difference of the given two numbers. His result was 4038 less than Petr's, who performed the calculation correctly.

Which two numbers could Adéla have written on the board? Determine all possibilities.

(L. Růžičková)
Let's denote the numbers written on the board as $a$ and $b$, assuming that $a \geq b$. From the problem statement, we can derive the following:

\[
\begin{aligned}
(a-b)^{2} & =a^{2}-b^{2}-4038, \\
a^{2}-2 a b+b^{2} & =a^{2}-b^{2}-4038, \\
2 a b-2 b^{2} & =4038, \\
b(a-b) & =2019,
\end{aligned}
\]

where the numbers $b$ and $a-b$ are positive according to the assumption.

The prime factorization of 2019 is $3 \times 673$. Therefore, the number 2019 can be expressed as the product of two positive integers in the following ways:

\[
2019=1 \cdot 2019=2019 \cdot 1=3 \cdot 673=673 \cdot 3.
\]

We emphasize the order of the factors to account for all possible assignments of $b$ and $a-b$. The corresponding pairs of numbers $(a, b)$ are $(2020,1)$, $(2020,2019)$, $(676,3)$, and $(676,673)$.

Scoring: 1 point for setting up the initial equation; 2 points for transforming it into a product form; 1 point for the prime factorization; 2 points for determining the valid pairs.

For a different solution approach, score 2 points for determining the valid pairs and 4 points for the quality of the commentary, especially the justification that no more solutions exist.