已知三个正实数x y z,且x+y+z=1,证明(x^2+y^2+z^2)(z/(x+y)+x/(y+z)+y/(z+x))>=1/2
问题描述：
已知三个正实数x y z,且x+y+z=1,证明(x^2+y^2+z^2)(z/(x+y)+x/(y+z)+y/(z+x))>=1/2
答
x²+y²+z²≥xy+yz+zx=1/2[x(y+z)+y(x+z)+z(x+y)]
所以(x^2+y^2+z^2)(z/(x+y)+x/(y+z)+y/(z+x))≥1/2[x(y+z)+y(x+z)+z(x+y)](z/(x+y)+x/(y+z)+y/(z+x））≥1/2(x+y+z)²=1/2
证毕
注：x²+y²+z²≥xy+yz+zx这一步应该懂吧
1/2[x(y+z)+y(x+z)+z(x+y)](z/(x+y)+x/(y+z)+y/(z+x））≥1/2(x+y+z)²这一步用的是柯西不等式.