3. (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 $7,9,10,10,11,13$. How many mushrooms did each collect?
Answer: $3,4,6,7$.

Solution. Let $x_{1} \leq x_{2} \leq x_{3} \leq x_{4}$ be the number of mushrooms collected by the friends. Then $x_{1}+x_{2}=7, x_{1}+x_{3}=9, x_{2}+x_{3}=10$. From this, $x_{1}=3, x_{2}=4, x_{3}=6$. Therefore, $x_{4}=13-x_{3}=7$. Since there are unused conditions, a check is necessary.

Remark. The problem can be solved without introducing literal unknowns. For example, 7+9+10 is the doubled quantity of mushrooms collected by the first three. Therefore, the three of them together collected 13, 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 (checking that the answer satisfies all six conditions) - minus 3 points (i.e., 9 points for the correct solution).