|
|
山川浮云
发表于 2024-7-20 20:28
本帖最后由 山川浮云 于 2024-7-20 22:47 编辑 令\(x={\tan }^{2}\alpha ,a={\tan }^{2}\beta ,\frac{8}{ax}={\tan }^{2}\gamma ,\alpha ,\beta ,\gamma \in (0,\frac{\pi }{2}),\)
问题转化为若\({\tan }^{2}\alpha {\tan }^{2}\beta {\tan }^{2}\gamma =8,\alpha ,\beta ,\gamma \in (0,\frac{\pi }{2}),\)则\(1<\cos \alpha +\cos \beta +\cos \gamma <2\).
已知条件即为\({\sin }^{2}\alpha {\sin }^{2}\beta {\sin }^{2}\gamma =8{\cos }^{2}\alpha {\cos }^{2}\beta {\cos }^{2}\gamma \)，\(\alpha ,\beta ,\gamma \in (0,\frac{\pi }{2}),\)
假设\(\cos \alpha +\cos \beta +\cos \gamma \leqslant 1 \),则\({\sin }^{2}\alpha=1-\cos^{2}\alpha>1-\cos \alpha\geqslant \cos\beta+\cos\gamma \geqslant2\sqrt{\cos\beta\cos\gamma}>2\cos\beta\cos\gamma \)
同理有\(\sin^{2}\beta>2\cos\gamma\cos\alpha, \ \ \ \ \sin^{2}\gamma>2\cos\alpha\cos\beta. \)
三式相乘得到\(\ \ \ \\{\sin }^{2}\alpha {\sin }^{2}\beta {\sin }^{2}\gamma >8{\cos }^{2}\alpha {\cos }^{2}\beta {\cos }^{2}\gamma. \)与已知条件矛盾.因此\(\cos \alpha +\cos \beta +\cos \gamma > 1. \)
假设\(\cos \alpha +\cos \beta +\cos \gamma \geqslant 2, \)则\({\sin }^{2}\alpha=(1+\cos\alpha)(1-\cos \alpha)<2(1-\cos\alpha)\leqslant 2(\cos\beta+\cos\gamma-1)<2\cos\beta\cos\gamma \)
同样方法可得到\(\ \ \ \\{\sin }^{2}\alpha {\sin }^{2}\beta {\sin }^{2}\gamma <8{\cos }^{2}\alpha {\cos }^{2}\beta {\cos }^{2}\gamma. \)又产生矛盾.
综上，结论成立.
参考08年江西最后一道压轴题的比较简洁的解法 |
|
kuing
发表于 2014-2-14 20:57
