2005 CMO 第 2 题

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$ . Prove that, there exist $x\in \mathbb{R}$ , satisfying two inequalities $$

\begin{aligned} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{aligned}

$$

if and only if $$\sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i).$$

假设 $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$ 。证明，存在 $x\in \mathbb{R}$ ，满足两个不等式 $$

\begin{对齐} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{对齐}

$$

当且仅当$$\sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i)。$$