- 高一数学难题，高手来当且仅当实数a,b,c满足什么条件时，不等式
- 当且仅当实数a,b,c满足什么条件时，不等式a(x-y)(x-z)+b(y-z)(y-x)+c(z-x)(z-y)≥0对任何x,y,z都成立？
- 此题为1988年IMO队选拔试题
a(x-y)(x-z)+b(y-z)(y-x)+c(z-x)(z-y)>=0对任何x,y,z都成立
令y=z,a(x-y)^2>=0,a>=0,同理:b>=0,c>=0
a(x-y)(x-z)+b(y-z)(y-x)+c(z-x)(z-y)
=a(x^2-xy-xz+yz)-b(x-y)(y-z)+c(z^2-xz-yz+xy)
=a(x^2-2xy+y^2)+a(xy-y^2-xz+yz)-b(x-y)(y-z)+c(y^2-2yz+z^2)+c(yz-y^2-xz+xy)
=a(x-y)^2+a(x-y)(y-z)-b(x-y)(y-z)+c(y-z)^2+c(x-y)(y-z)
=a(x-y)^2+(a-b+c)(x-y)(y-z)+c(y-z)^2>=0
上式为(x-y),(y-z)的二次齐次式,a>=0,值为非负的充要条件:
△=(a-b+c)^2-4ac<=0,
a^2+b^2+c^2<=2(ab+bc+ca)
故:a,b,c应满足:
1)a>=0,b>=0,c>=0
2)a^2+b^2+c^2<=2(ab+bc+ca)