Question

设a、b、c是不全相等的任意实数,若x=a^2-bc,y=b^2-ca,z=c^2-ab,求证：x,y,z中至少有一个大

Answer 1

x+y+z=a^2+b^2+c^2-2ab-2bc-2ca

2(x+y+z)=2a^2+2b^2+2c^2-2ab-2bc-2ca

2(x+y+z)=(a^2-2ab+b^2)+(b^2-2bc+c^2)+(c^2-2ca+a^2)

2(x+y+z)=(a-b)^2+(b-c)^2+(c-a)^2>0

所以x、y、z中至少有1个大于0