MathDB

1

Part of 2005 China National Olympiad

Problems(1)

trigonometric inequalities from china 2005

Source: china mathematical olympiad cmo 2005 final round - Problem 1

1/22/2005
Suppose θi∈(−π2,π2),i=1,2,3,4\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4. Prove that, there exist x∈Rx\in \mathbb{R}, satisfying two inequalities \begin{eqnarray*} \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{eqnarray*} if and only if ∑i=14sin⁡2θi≤2(1+∏i=14sin⁡θi+∏i=14cos⁡θi). \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i).
inequalitiestrigonometryinequalities unsolved