AIME 1987 P5

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Problem Statement

Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.

Problem Link

Solution

$$y^2 + 3x^2y^2 - 30x^2 = 517$$ $$y^2 + 3x^2(y^2 - 10) = 517$$ $$y^2 - 10 + 3x^2(y^2 - 10) = 517 - 10$$ $$(3x^2 + 1)(y^2 - 10) = 507 = 1 \times 507 = 3 \times 169 = 13 \times 39$$ $$\text{The only solution that works is since } x, y \in \mathbb{Z} \text{ is } y^2 - 10 = 39, 3x^2 + 1 = 13$$ $$\therefore 3x^2 = 12, y^2 = 49, 3x^2y^2 = 12 \times 49 = 588$$