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Putnam 1973 B6

Source: Putnam 1973

May 25, 2022
Putnamalgebrafunctiondomaintrigonometryinequalities

Problem Statement

On the domain 0≤θ≤2π:0\leq \theta \leq 2\pi: (a) Prove that sin⁡2θ⋅sin⁡2θ\sin^{2}\theta \cdot \sin 2\theta takes its maximum at π3\frac{\pi}{3} and 4π3\frac{4 \pi}{3} (and hence its minimum at 2π3\frac{2 \pi}{3} and 5π3\frac{ 5 \pi}{3}). (b) Show that ∣sin⁡2θ⋅sin⁡32θ⋅sin⁡34θ⋯sin⁡32n−1θ⋅sin⁡2nθ∣| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta | takes its maximum at 4π3\frac{4 \pi}{3} (the maximum may also be attained at other points). (c) Derive the inequality: sin⁡2θ⋅sin⁡22θ⋯sin⁡22nθ≤(34)n. \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.
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