1. In the set of real numbers, solve the equation

$$
x=\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}
$$
The given equation is equivalent to the system

$$
\left(x-\sqrt{x-\frac{1}{x}}\right)^{2}=x-\frac{1}{x}, \quad x \geq 1
$$

or to the system

$$
x^{2}-x-1=0, \quad x \geq 1
$$

The only solution to the last system is $x=\frac{1+\sqrt{5}}{2}$.