A set of several numbers, none of which are the same, has the following property: the arithmetic mean of some two numbers from this set is equal to the arithmetic mean of some three numbers from the set and is equal to the arithmetic mean of some four numbers from the set. What is the smallest possible number of numbers in such a set? 

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Let $C\left(a_{1}, \ldots, a_{k}\right)$ be the arithmetic mean of the numbers $\left(a_{1}, \ldots, a_{k}\right)$. Note that adding a number to the set that is different from its arithmetic mean changes the original arithmetic mean of the set.

Suppose that $(a, b, c, d)$ is a set of four numbers satisfying the condition, and $C(a, b, c, d)=C(a, b, c)=$

C. Then $d=C$.

A set of two different numbers with the same arithmetic mean cannot contain the number $d$. If, for example, these are the numbers $a$ and $b$, then $c=C=d$, which contradicts the condition.

Therefore, a set of four numbers satisfying the condition does not exist.

An example of a set of five numbers: $1,2,3,4,5$. Indeed, $C(2,4)=C(2,3,4)=C(1,2,4,5)=3$.

## Answer

5 numbers.