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AIME 2025 I / The parabola with equation y=x²−4 is rotated 60^(∘) counterclockwise…

Problem

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question

The parabola with equation

y=x^{2}-4

is rotated

60^{\circ}

counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has

y

-coordinate

\frac{a-\sqrt{b}}{c}

, where

a

b

, and

c

are positive integers, and

a

and

c

are relatively prime. Find

a+b+c

Plain-text mathematical notation (without MathML)

The parabola with equation y=x²−4 is rotated 60^(∘) counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has y-coordinate (a−√(b))/(c), where a, b, and c are positive integers, and a and c are relatively prime. Find a+b+c.

Original LaTeX notation

The parabola with equation $y=x^{2}-4$ is rotated $60^{\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a+b+c$.

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Source and history

task_id: eda8c6ff-6dc7-5873-b09e-4a235b512b82
task_key: AIME2025~2dI--test--eda8c6ff-6dc7-5873-b09e-4a235b512b82
task_revision_id: 2
config / split: AIME2025-I / test

Official source

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