<s>[INST] Inequality in sum of fractions 
Is there a constant $C\ge 0$ such that for any $a,b,c,d>0$ the inequality holds:
  $$
\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \ge C+\frac{a+b}{b+c}+\frac{b+c}{c+d}+\frac{c+d}{d+a}
$$


My attempt:
$a=b=c=d$ gives $1\ge C$.
Now, take $b=c=d$; this leads to inequality with two variables ($a$ and, let's say, $b$), which for $C=1$ is equivalent to $a^3+2b^3\ge 3ab^2$. This one is true by Cauchy.
The case $a=b$, $c=d$ leads to the same thing.
Is $C=1$ good?
 [/INST] For $C=1$ it's wrong. Try $a=10$, $b=15$, $c=10$ and $d=11$.
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