Interesting system of equations

[ins-]

Solve (in reals) the system:
$x+y=\frac{1}{x}+\frac{1}{y}+1$
$y+z=\frac{1}{y}+\frac{1}{z}+2$
$z+x=\frac{1}{z}+\frac{1}{x}+3$

[NSKuber]

$a = x - \frac{1}{x}, b = y - \frac{1}{y}, c = z - \frac{1}{z}$

$a + b = 1$
$b + c = 2$
$c + a = 3$

$a=1, b=0, c=2$
$x = \frac{1\pm\sqrt{5}}{2}, y = \pm 1, z = 1 \pm \sqrt{2}$

[ins-]

Using this approach also this one can be solved:
$x+y=\frac{1}{x}-\frac{1}{y}+1$
$y+z=\frac{2}{y}+\frac{1}{z}+2$
$z+x=\frac{1}{z}+\frac{1}{x}+3$