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广州王老师(5234****) 16:03:43
这题怎么证？
设
\[t=\frac{1+\sqrt{33}}{8}\approx 0.843, \]
下面证明更强的命题
\[\min \{a_{1},a_{2},a_{3}\}\le t, \]
令
\[a_{1}=t+x,a_{2}=t+y,a_{3}=t+z
\riff x+y+z=\frac{9}{2}-3t,\]
则
\[2=(t+x)(t+y)(t+z)=t^{3}+t^{2}\left( \frac{9}{2}-3t \right)+t(xy+yz+zx)+xyz, \]
由 $t$ 的具体数值不难验证有
\[t^{3}+t^{2}\left( \frac{9}{2}-3t \right)=2, \]
因此有
\[t(xy+yz+zx)+xyz=0, \]
由此可见
\[\min \{x,y,z\}\le 0 \riff\min \{a_{1},a_{2},a_{3}\}\le t. \] |
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