kill(all)$

van=facebook.com/LearningCalculus$

f(x,y,z):=((x^2+y)*(x+y^2))/(x+y-1)^2 + ((y^2+z)*(y+z^2))/(y+z-1)^2 +((z^2+x)*(z+x^2))/(z+x-1)^2 - 2*(x+y+z);

min:inf;

findT(n) := for x: -n thru n do
                for y: -n thru n do
                     for z: -n thru n do
                        if x + y # 1 and y + z # 1 and z + x # 1 then
                            (T:f(x,y,z),if T < min then min:T)$

findT(1)$ min;

findT(10)$ min;

findT(30)$ min;

float(%);