Question

非负实数X、Y、Z满足条件：XY+YZ+XZ=1,求证：1/(X+Y)+1/(Y+Z)+1/(X+Z)

Answer 1

证明：

设f(x,y,z)=1/(x+y)+1/(y+z)+1/(z+x)

由对称性不妨设0(1-xy)/(x+y)=z}

2+2z^2=t+1/t,而由均值不等式t>=2,而我们知道函数t+1/t当x>=1时是单调递增的.所以f(x,y,z)>=t+1/t>=5/2

证毕