证明1/2x+1/2y+1/2z≥1/(x+y)+1/(x+z)+1/(z+y)

Question

证明1/2x+1/2y+1/2z≥1/(x+y)+1/(x+z)+1/(z+y)

Answer 1

应该有条件：x,y,z都大于0
要证1/2x+1/2y+1/2z≥1/(x+y)+1/(x+z)+1/(z+y)
只需证(1/x+1/y)/4≥1/(x+y) (1/x+1/z)/4≥1/(x+z) (1/z+1/y)/4≥1/(z+y)
(x-y)^2≥0
x^2-2xy+y^2≥0
x^2+2xy+y^2≥4xy
(x+y)^2≥4xy
(x+y)/(xy)≥4/(x+y)
(1/x+1/y)/4≥1/(x+y)
同理可证(1/x+1/z)/4≥1/(x+z) (1/z+1/y)/4≥1/(z+y)
将三式相加就可得1/2x+1/2y+1/2z≥1/(x+y)+1/(x+z)+1/(z+y)