Prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}\geq1+4\sqrt{\frac{abcd}{(a+b)(b+c)(c+d)(d+a)}}$ Let $a$, $b$, $c$ and $d$ be positive numbers. Prove that:
$$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}\geq1+4\sqrt{\frac{abcd}{(a+b)(b+c)(c+d)(d+a)}}$$
We can prove this inequality by using $b=a+u$, $c=a+v$ and $d=a+w$ after squaring of the both sides.
I am looking for a nice proof. Thank you!

Here is an easy proof.
Let $x=\frac{b}{a}$, $y=\frac{c}{b}$, $z=\frac{d}{c}$ and $t=\frac{a}{d}$.
Hence, $xyzt=1$ and we need to prove that  $\sum\limits_{cyc}\frac{1}{x+1}\geq1+\frac{4}{\sqrt{\prod\limits_{cyc}(x+1)}}$.
Let $f(x,y,z,t,\lambda)=\sum\limits_{cyc}\frac{1}{x+1}-1-\frac{4}{\sqrt{\prod\limits_{cyc}(x+1)}}+\lambda(xyzt-1)$.
It's obvious that in the minimum point of $f$ happens $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=\frac{\partial f}{\partial t}=\frac{\partial f}{\partial \lambda}=0$.
But $\frac{\partial f}{\partial x}=-\frac{1}{(x+1)^2}+\frac{2}{(x+1)\sqrt{\prod\limits_{cyc}(x+1)}}+\frac{\lambda}{x}$, which gives $(x+1)\frac{\partial f}{\partial x}=-\frac{1}{x+1}+\frac{2}{\sqrt{\prod\limits_{cyc}(x+1)}}+\frac{\lambda}{x}+\lambda$.
Thus, $-\frac{1}{x+1}+\frac{\lambda}{x}=-\frac{1}{y+1}+\frac{\lambda}{y}$, which gives $(x-y)\left(\frac{xy}{(x+1)(y+1)}-\lambda\right)=0$. Similarly we obtain:
$(x-z)\left(\frac{xz}{(x+1)(z+1)}-\lambda\right)=0$, $(x-t)\left(\frac{xt}{(x+1)(t+1)}-\lambda\right)=0$, $(y-z)\left(\frac{yz}{(y+1)(z+1)}-\lambda\right)=0$,
$(y-t)\left(\frac{yt}{(y+1)(t+1)}-\lambda\right)=0$ and $(z-t)\left(\frac{zt}{(z+1)(t+1)}-\lambda\right)=0$.
Since $\frac{xy}{(x+1)(y+1)}-\lambda=\frac{xz}{(x+1)(z+1)}-\lambda\Leftrightarrow y=z$,
we see that in the minimum point or two variables are equal twice or three variables are equal.


*

*$y=x$ and $z=t=\frac{1}{x}$.


In this case $\sum\limits_{cyc}\frac{1}{x+1}\geq1+\frac{4}{\sqrt{\prod\limits_{cyc}(x+1)}}\Leftrightarrow(x+1)^2\geq4x$, which is obvious; 


*$y=z=x$ and $t=\frac{1}{x^3}$.


In this case $\sum\limits_{cyc}\frac{1}{x+1}\geq1+\frac{4}{\sqrt{\prod\limits_{cyc}(x+1)}}\Leftrightarrow(x-1)^2(9x^4+2x^3+5x^2+4x+4)$, which is obvious too.
Done!
