- 不等式问题题设x,y,z,w为正数。求证：(y+z+w)/x+(
- 题 设x,y,z,w为正数。求证：
(y+z+w)/x+(z+w+x)/y+(w+x+y)/z+(x+y+z)/w≧9[x/(y+z+w)+y/(z+w+x)+z/(w+x+y)+w/(x+y+z)
- 题 设x,y,z,w为正数。求证：
(y+z+w)/x+(z+w+x)/y+(w+x+y)/z+(x+y+z)/w≧9[x/(y+z+w)+y/(z+w+x)+z/(w+x+y)+w/(x+y+z)
证明: 因为 1/y+1/z+1/w>=9/(y+z+w)
所以 x/y+x/z+x/w>=9x/(y+z+w),
同理 y/z+y/w+y/x>=9y/(z+w+x),
z/w+z/x+z/y>=9z/(w+x+y),
w/x+w/y+w/z>=9w/(x+y+z),
上面四式叠加,即得
(y+z+w)/x+(z+w+x)/y+(w+x+y)/z+(x+y+z)/w≧9[x/(y+z+w)+y/(z+w+x)+z/(w+x+y)+w/(x+y+z)