# Problem 4.

When processing numerical data, it is often necessary to calculate the arithmetic mean

$$
S(x, y)=(x+y) / 2
$$

and solve equations containing the arithmetic mean. Find all finite (consisting of a finite number of elements) numerical sets $X$ such that for any $a$ and $b$ from $X$ the set $X$ contains the root $x$ of the equation

$$
S(a, x)=b
$$
Solution. We have

$$
S(a, x)=b \Leftrightarrow x=2 b-a
$$

The required property in the problem is possessed by any one-element set

$$
X=\{a\}, \quad a \in(-\infty ; \infty)
$$

since $S(a, a)=a$.

Suppose further that the set $X$ contains at least two distinct elements $c, d$, with $c<d$ (without loss of generality). For the equation $S(d, x)=c$, we find, according to (1), $x=2 c-d$. Then for the equation $S(d, x)=2 c-d$ we get $x=4 c-3 d$, and then we consider the equation $S(d, x)=4 c-3 d$ and get $x=16 c-15 d$. Continuing in this manner, we obtain the sequence of solutions

$$
c, 2 c-d, 4 c-3 d, 16 c-15 d, \ldots
$$

We will show that all its terms $x_{n}=2^{n} c-\left(2^{n}-1\right) d, \quad n=0,1,2, \ldots$, are pairwise distinct. If we assume that $x_{n}=x_{m}$ for $n \neq m$, then, transforming the equality, we get $\left(2^{m}-2^{n}\right) c=\left(2^{m}-2^{n}\right) d$, from which $c=d$, which is impossible. Thus, the set $X$ contains an infinite subset - the sequence (3), hence the set $X$ is infinite.

Answer: exactly all one-element sets $X=\{a\}, a \in(-\infty ; \infty)$.