9.1. Four non-zero numbers are written on the board, and the sum of any three of them is less than the fourth number. What is the smallest number of negative numbers that can be written on the board? Justify your answer.
Solution: Let the numbers on the board be $a \geqslant b \geqslant c \geqslant d$. The condition of the problem is equivalent to the inequality $a+b+c < d$ for optimality | 7 points |
| There is a proof that there are no fewer than three negative numbers (in the absence of an example with three numbers) | 3 points |
| An example of a set with three negative numbers is provided (in the absence of a proof of its optimality) | 1 point |
| Correct answer without justification (or with incorrect justification) | 0 points |
| Examples of sets with four negative numbers | not evaluated |