2023 AMC 10B Problem 14

Below is the professionally curated solution for Problem 14 of the 2023 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10B solutions, or check the answer key.

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Concepts:Diophantine Equation bounding to limit cases casework

Difficulty rating: 1660

14.

How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2 + mn + n^2 = m^2 n^2?$

$7$

$1$

$3$

$6$

$5$

Solution:

If $m = 0,$ the equation forces $n^2 = 0,$ giving $(0, 0).$ Otherwise both are nonzero; assume $|m| \le |n|.$ Then $m^2 n^2 = m^2 + mn + n^2 \le 3n^2,$ so $m^2 \le 3$ and $m = \pm 1.$ Take $m = 1:$ $1 + n + n^2 = n^2$ gives $n = -1.$ Take $m = -1:$ $n = 1.$ That leaves $(0,0), (1,-1), (-1,1),$ three in all. Therefore, the answer is C.

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