1. (12 points) Four friends went to the forest to pick mushrooms. Upon returning, every pair of them counted the total number of mushrooms they had collected. The numbers obtained were $6,7,9,9,11,12$. How many mushrooms did each collect?
Answer: $2,4,5,7$.

Solution. Let $x_{1} \leq x_{2} \leq x_{3} \leq x_{4}$ be the quantities of mushrooms collected by the friends. Then $x_{1}+x_{2}=6, x_{1}+x_{3}=7, x_{2}+x_{3}=9$. From this, $x_{1}=2, x_{2}=4, x_{3}=5$. Therefore, $x_{4}=12-x_{3}=7$. Since there are unused conditions, a check must be performed.

Remark. The problem can be solved without introducing literal unknowns. For example, $6+7+9$ is twice the number of mushrooms collected by the first three. Therefore, the three of them together collected 11, and so on.

Grading criteria. Full solution: 12 points. Correct answer obtained, for example, by enumeration, without proof of uniqueness: 4 points. Guessed answer plus proof of existence (verification) and uniqueness: 12 points. Lack of verification (determining that the answer satisfies all six conditions) - minus 3 points (i.e., 9 points for a correct solution).