HMMT February

Since $\lfloor x\rfloor>x-1$ for all $x$, we have that $$\begin{aligned} \left\lfloor\frac{a+b}{c}\right\rfloor+\left\lfloor\frac{b+c}{a}\right\rfloor+\left\lfloor\frac{c+a}{b}\right\rfloor & >\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}-3 \\ & =\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)-3 \end{aligned}$$ But by the AM-GM inequality, each of the first three terms in the last line is at least 2. Therefore, the lefthand side is greater than $2+2+2-3=3$. Since it is an integer, the smallest value it can be is 4. This is in fact attainable by letting $(a, b, c)=(6,8,9)$.