On the table lay five cards with different positive integers. Mark calculated that the largest of the numbers on the cards is 8 greater than the smallest number. Adam took the cards with the two smallest numbers and calculated that the product of these two numbers is 12. Dominik then calculated that the sum of the numbers on the remaining three cards is 25.

Which numbers could have been written on the cards? Find three solutions.

(L. Růžičková)
The number 12 can be obtained as a product of two positive integers, where the first is smaller than the second, in three ways:

$$
1 \cdot 12 = 2 \cdot 6 = 3 \cdot 4 \text{.}
$$

If the smallest two numbers were 1 and 12, the largest number would be $1 + 8 = 9$. However, this is not possible because 12 is not less than 9.

If the smallest two numbers were 2 and 6, the largest number would be $2 + 8 = 10$. The remaining two numbers would then be different numbers greater than 6 and less than 10, which together with the number 10 give a sum of 25. This is satisfied only by the numbers 7 and 8.

If the smallest two numbers were 3 and 4, the largest number would be $3 + 8 = 11$. The remaining two numbers would then be different numbers greater than 4 and less than 11, which together with the number 11 give a sum of 25. This is satisfied by the pairs of numbers 5 and 9, as well as 6 and 8.

The three sets of five numbers that satisfy the conditions are

$$
(2, 6, 7, 8, 10), \quad (3, 4, 5, 9, 11), \quad (3, 4, 6, 8, 11)
$$

Scoring. 2 points for each set of five numbers: 1 point for finding the set and 1 point for deriving or verifying that it meets the conditions.