9.1. A finite set of distinct real numbers $X$ is called good if each number in $X$ can be represented as the sum of two other distinct numbers in $X$. What is the minimum number of elements that a good set $X$ can contain?

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# Answer: 6.

Solution. From the condition, it follows that $X$ contains no less than three numbers, which means there are non-zero numbers in it. By multiplying all numbers by minus one if necessary, we can assume that $X$ contains positive numbers. Let's choose the largest number $M$ from them. According to the condition, it is equal to the sum of two other distinct numbers from $X$, each of which is smaller than it. From the maximality of $M$, it follows that both of these numbers are positive. Therefore, $X$ contains no less than three distinct positive numbers. Consider the smallest positive number $\boldsymbol{m}$, it is equal to the sum of two other distinct numbers from $X$, one of which must be negative due to the minimality of $\boldsymbol{m}$. Now let's choose the smallest negative number $\mathrm{N}$ from $X$, it is also equal to the sum of two other distinct numbers from $X$, each of which is greater than it. From the minimality of $\mathrm{N}$, it follows that both of these numbers are negative. Therefore, $X$ contains no less than three distinct negative numbers. Thus, a good set $X$ must contain no fewer than six numbers.

An example of a good set with six elements is the set $X=\{-3,-2,-1,1,2,3\}$.

Grading Criteria. ($\cdot$) An example of a good set with six numbers is provided: 2 points. ($\cdot$) It is proven that $X$ contains no less than 3 positive or negative numbers: 3 points. ($\cdot$) It is proven that $X$ must contain numbers of different signs: 2 points.