1987 AIME Problems/Problem 5

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Problem

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

Solution

If we move the $x^2$ term to the left side, it is factorable with Simon's Favorite Factoring Trick:

\[(3x^2 + 1)(y^2 - 10) = 517 - 10\]

$507$ is equal to $3 \cdot 13^2$. Since $x$ and $y$ are integers, $3x^2 + 1$ cannot equal a multiple of three. $169$ doesn't work either, so $3x^2 + 1 = 13$, and $x^2 = 4$. This leaves $y^2 - 10 = 39$, so $y^2 = 49$. Thus, $3x^2 y^2 = 3 \times 4 \times 49 = \boxed{588}$.

Video Solution by OmegaLearn

https://youtu.be/ba6w1OhXqOQ?t=4699 ~ pi_is_3.14

Video Solution

https://youtu.be/z4-bFo2D3TU?list=PLZ6lgLajy7SZ4MsF6ytXTrVOheuGNnsqn&t=3704 - AMBRIGGS

See also

1987 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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All AIME Problems and Solutions

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