<s>[INST] Advanced Inequality with Floor Functions For positive real numbers $a,$ $b,$ $c,$ and $d,$ find the minimum value of
$$\left\lfloor \frac{b + c + d}{a} \right\rfloor + \left\lfloor \frac{a + c + d}{b} \right\rfloor + \left\lfloor \frac{a + b + d}{c} \right\rfloor + \left\lfloor \frac{a + b + c}{d} \right\rfloor.$$
I'm not seeing a technique to easily solve this problem; at this moment, I'm thinking a property of some basic inequalities such as Cauchy or AM-GM will help. Other than that, I really have no idea. Solutions?
 [/INST] By C-S $$\sum_{cyc}\left[\frac{a+b+c}{d}\right]>\sum_{cyc}\frac{a+b+c}{d}-4=\sum_{cyc}a\sum_{cyc}\frac{1}{a}-8\geq16-8=8.$$
Now, make an example for $$\sum_{cyc}\left[\frac{a+b+c}{d}\right]=9.$$
I got that $(a,b,c,d)=(5,5,5,4)$ is valid.
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