- 高中竞赛题设x,y,z>0,求证x/(y^2+z^2)+y
- 设x,y,z>0,求证
x/(y^2+z^2)+y/(z^2+x^2)+z/(x^2+y^2)>4/(x+y+z)
- 设x,y,z>0,求证
x/(y^2+z^2)+y/(z^2+x^2)+z/(x^2+y^2)>4/(x+y+z) (1)
先证一个不等式:
∑x^2*∑(yz)^2>∏(y^2+z^2) (2)
(2)式展开为: (xyz)^2>0
下面给出(1)式加强式:
∑x/(y^2+z^2)≥4∑x^2*∑(yz)^2/[∑x*∏(y^2+z^2) (3)
(3)＜＝＝＞
∑x^6+2∑(y+z)x^5-3∑(y^2+z^2)x^4+2xyz∑x^3
+xyz∑(y+z)x^2-9(xyz)^2≥0 (4)
设x=min(x,y,z),(4)分解化简为
x[x^3+3(y+z)x^2+7xyz+5yz(y+z)](x-y)(x-z)
+[y^4+z^4+4yz(y^2+z^2+yz)+2x(y+z)^3-3(y^2+z^2)x^2](y-z)^2≥0
上式为显然.