Maths Olympiad Prep

Library / /25 of 520

Algebra Difficulty 5.6 AIME, harder Find the answer

Example 1 Let x,y,zx, y, z be real numbers, 0<x,y,z<π0 < x, y, z < \pi. Find the maximum value of sin⁡2x+sin⁡2y+sin⁡2z\sin 2 x+\sin 2 y+\sin 2 z (1990 China National

A number or a short expression. Spacing and $ signs are ignored.

Solution

Prove that the original inequality can be transformed into
π4>sin⁡x(cos⁡x−cos⁡y)+sin⁡y(cos⁡y−cos⁡z)+sin⁡zcos⁡z\begin{array}{l} \frac{\pi}{4}>\sin x(\cos x-\cos y)+ \\ \sin y(\cos y-\cos z)+\sin z \cos z \end{array}

Construct a figure as shown in Example 1, where circle
OO is a unit circle,
S1=sin⁡x(cos⁡x−cos⁡y)S2=sin⁡y(cos⁡y−cos⁡z)\begin{array}{l} S_{1}=\sin x(\cos x-\cos y) \\ S_{2}=\sin y(\cos y-\cos z) \end{array}
S3=sin⁡zcos⁡zS_{3}=\sin z \cos z

Since S1+S2+S3=sin⁡x(cos⁡x−cos⁡y)+sin⁡y(cos⁡y−cos⁡z)+sin⁡zcos⁡z S_{1}+S_{2}+S_{3} = \sin x(\cos x-\cos y) + \sin y(\cos y-\cos z) + \sin z \cos z .

Therefore, the original inequality holds.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.