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Annual Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:

$\alpha_{1}, \alpha_{2}, \alpha_{3}$ , and $\alpha_{4}$ are the complex roots of the equation $x^{4}+2 x^{3}+2=0$ . Determine the unordered set
$\left\{\alpha_{1} \alpha_{2}+\alpha_{3} \alpha_{4}, \alpha_{1} \alpha_{3}+\alpha_{2} \alpha_{4}, \alpha_{1} \alpha_{4}+\alpha_{2} \alpha_{3}\right\}$

Solution

Solution:

$\{1 \pm \sqrt{5},\ -2\}$ . Same as Algebra $\# 9$ .

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