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(2a-1) sin x + (1-a) sin(1-a)x >= 0 for 0 <=a <= 1$ and 0 <= x <= \pi

Source: 1982 Swedish Mathematical Competition p6

March 28, 2021
trigonometryinequalitiesalgebra

Problem Statement

Show that (2a−1)sin⁡x+(1−a)sin⁡(1−a)x≥0 (2a-1) \sin x + (1-a) \sin(1-a)x \geq 0 for 0≤a≤10 \leq a \leq 1 and 0≤x≤π0 \leq x \leq \pi.
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