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

题目

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

答案

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≥0x^2-2xy+y^2≥0x^2+2xy+y^2≥4xy(x+y)^2≥4xy(x+y)/(xy)≥4...