Fabián has four cards, on each of which he wrote a different positive integer less than 10. He wrote the numbers in different colors, with the following conditions:

- The product of the green and yellow numbers is the green number.
- The blue number is the same as the red number.
- The product of the red and blue numbers is a two-digit number written with the green and yellow digits (in that order).

Determine these four numbers.

(M. Petrová)
Fabián writes only natural numbers. From the first condition, it follows that if we multiply the green number by the yellow one, we do not change its value. This means that the yellow number is 1. From the second condition, it follows that the blue and red numbers he wrote are the same. From the third condition, it then follows that if we multiply two red (or two blue) numbers, we get a two-digit number ending in one (the yellow number is 1). Let's go through all the possibilities:

$$
\begin{gathered}
1 \cdot 1=1, \quad 2 \cdot 2=4, \quad 3 \cdot 3=9, \quad 4 \cdot 4=16 \\
5 \cdot 5=25, \quad 6 \cdot 6=36, \quad 7 \cdot 7=49, \quad 8 \cdot 8=64, \quad 9 \cdot 9=81 .
\end{gathered}
$$

The given conditions are satisfied by only one possibility, and that is the last one. Fabián wrote the number 9 in red, 9 in blue, 8 in green, and 1 in yellow.

Scoring suggestion. 2 points for determining the yellow number, with 1 point for the justification; 3 points for examining all possibilities and finding the product $9 \cdot 9=81$; 1 point for correctly determining the other three numbers.

If the solver does not list all possible products (they only have the final product $9 \cdot 9=81$), does not justify that there are no more solutions, and does not explain that they have examined all possibilities and found no other solutions, award no more than 4 points.

The assignment of colors to individual numbers is not a necessary part of the solution, so it should not be scored in any way.