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HMMT November

Algebra Difficulty 5.0 AIME Find the answer

Given an angle θ\theta, consider the polynomial P(x)=sin⁡(θ)x2+cos⁡(θ)x+tan⁡(θ)x+1P(x)=\sin(\theta)x^{2}+\cos(\theta)x+\tan(\theta)x+1. Given that PP only has one real root, find all possible values of sin⁡(θ)\sin(\theta).

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that if sin⁡(θ)=0\sin(\theta)=0, then the polynomial has 1 root. Now assume this is not the case then the polynomial is a quadratic in xx. Factor the polynomial as (tan⁡(θ)x+1)(x+sec⁡(θ))(\tan(\theta)x+1)(x+\sec(\theta)). Then the condition is equivalent to sec⁡(θ)=1tan⁡(θ)\sec(\theta)=\frac{1}{\tan(\theta)}, which is equivalent to sin⁡(θ)=cos⁡2(θ)=1−sin⁡2(θ)\sin(\theta)=\cos^{2}(\theta)=1-\sin^{2}(\theta). Solving now gives sin⁡(θ)=5−12\sin(\theta)=\frac{\sqrt{5}-1}{2} as the only solution.

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