5. What is the smallest number of different numbers that can be chosen so that each chosen number is equal to the sum of some three other different chosen numbers?

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Answer. Seven. Solution. Example. $-3,-2,-1,0,1,2,3$. Evaluation. Let $n-$ be the largest of the chosen numbers. If $n \leq 0$, all other chosen numbers are negative, and the sum of any two of them is less than each of the addends, and therefore cannot equal $n$. Therefore, $n>0$. The two of the chosen numbers whose sum equals $n$ must also be positive, otherwise their sum would be less than $n$. Thus, there must be at least three positive numbers among the chosen ones. Considering the smallest of the chosen numbers, similarly, we can conclude that there must be at least three negative numbers among the chosen ones. Therefore, a total of at least six numbers must be chosen.

Suppose exactly six numbers are chosen: $a<b<c<d<e<f$. From the above, it follows that $c<0<d$. By changing the signs of all numbers to their opposites if necessary, we can assume that $d \leq|c|$. But then the number $f$ cannot be represented as the sum of three other chosen ones, since even $c+d+e \leq e<f$.