An amazing system

[ins-]

Solve the system:
$\frac{1}{x+y}+x+y+z=2$
$\frac{2}{y+z}+x+y+z=3$
$\frac{3}{z+x}+x+y+z=4$

(Оффтоп)

Luckily it is solvable

[waxtep]

If we take $s=x+y+z$, a very nice equation for $s$ emerges:
$\frac s {s-1}=\frac {s-1} {s-2}+\frac {s-2}{s-3}+\frac {s-3}{s-4}$

-- 28.04.2016, 01:20 --

A-ha, and then if we proceed accurately, we get a good возвратное equation $t^4-t^3-t^2-t+1=0$ for $t=s-2$, which solves the system!

[ins-]

Another approach is to multiply all the equations by 2. Then to substitute: $u=x+y$, $v=y+z$, $w=z+x$. If we denote the resulting equation by 1), 2), 3). After we subtract 2)-1) we have a dependency between $u$ and $v$ *). After we subtract 3)-1) we have a dependency between $w$ and $u$ **). Replacing $v$ and $w$ from *) and **) in 1) will give us the equation: $u^4+u^3-u^2+u+1=0$, by dividing it to $u^2$ and substituting $k=u+\frac{1}{u}$ we have a square equation for $k$. The rest is easy.

[ins-]

One more solution: http://artofproblemsolving.com/communit ... 64p6253448 . It can be solved in more ways. The common between them is the reciprocal equation.