<s>[INST] the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $ If we know that for every $a,b,c>0$ ,how we can find  the least possible value for :
$$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $$
 [/INST] Put $(a,b,c)=(3,4,4)$ to get 4. I will show this is optimal.
Assume without loss of generality that $a \leq b \leq c$. Two cases:
If $c \geq a+b$ then $\lfloor\frac{c+a}{b}\rfloor \geq 1$ and $\lfloor\frac{c+b}{a} \rfloor \geq \lfloor\frac{a+b+b}{a} \rfloor \geq 3$ so the sum is at least 4.
If $c \leq a+b$ then $\lfloor\frac{a+b}{c}\rfloor \geq 1$, $\lfloor\frac{b+c}{a}\rfloor \geq 2$, $\lfloor\frac{c+a}{b}\rfloor \geq 1$ so the sum is at least 4.
</s>