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HMMT November, 2022

Algebra Difficulty 5.0 AIME Prove it United States

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

Solution

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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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.