Minimum value of a floor function

Question

Minimum value of a floor function. $\left \lfloor{(a+b+c)/d}\right \rfloor + \left \lfloor{(a+b+d)/c}\right \rfloor +\left \lfloor{(a+d+c)/b}\right \rfloor + \left \lfloor{(d+b+c)/a}\right \rfloor$ My working: For a=b= -d= -c, I'm getting -4 as the answer. Is there a concrete method to find the minimum value?

Answer 1

If a,b,c,d are positive: By $\lfloor x\rfloor>x-1$, the function $f= \lfloor{(a+b+c)/d} \rfloor + \lfloor{(a+b+d)/c} \rfloor + \lfloor{(a+d+c)/b} \rfloor + \lfloor{(d+b+c)/a} \rfloor$$>(a+b+c)/d+(a+b+d)/c+(a+d+c)/b+(d+b+c)/a-4=(a+b+c+d)/d+(a+b+c+d)/c+(a+b+c+d)/b+(a+b+c+d)/a-8$$=(a+b+c+d)(1/a+1/b+1/c+1/d)-8$ (By Cauchy-Schwarz) $\ge (a\cdot\frac{1}{a}+b\cdot\frac{1}{b}+c\cdot\frac{1}{c}+d\cdot\frac{1}{d})^2-8=8$.

As an integer, $f>8$ means $f\ge 9$. When a=b=c=1.1 and d=1, the function f=9, hence the Minimum value of the floor function is 9.

If we allow a,b,c,d to be negative: Let $a=b=c=-1,d=\frac{1}{n}$ where $n\ge 2 $ is any positive integer, then $f=3-3n$ In this case, $f$ can be sufficiently small.