# Problem 4.

When designing a certain technical device, there arose a need to solve equations

$$
a \circ x=b,
$$

where the operation $\circ$ over two numbers is defined by the condition

$$
y \circ z=\frac{y+z+|y-z|}{2} \text {. }
$$

Find all numerical sets $X$ such that for any $a, b$ from $X$ the specified equation has a unique solution $x$ and this solution belongs to the set $X$.
# Solution.

Notice that

$$
y \circ z=\max \{y, z\}
$$

Next, any one-element set fits

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

since $\max \{a, a\}=a$.

Suppose that the set $X$ contains at least two distinct elements $a$ and $b$. Without loss of generality, assume $a>b$. For such $a$ and $b$, equation (1) has no solution, since $\max \{a, x\} \geq a$.

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