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Philippines Mathematical Olympiad

Algebra Difficulty 5.2 AIME, harder Prove it Philippines

Problem:

For what real values of $k$ does the system of equations
$\left\{\begin{array}{l} x - k y = 0 \\ x^{2} + y = -1 \end{array}\right.$
have real solutions?

Solution

Solution:

$-\frac{1}{2} \leq k \leq \frac{1}{2}$

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