2020 AMC 12B Problem 8

Attempt Problem 8 of the 2020 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

How many ordered pairs of integers $(x, y)$ satisfy the equation

$x^{2020} + y^2 = 2y?$

$1$

$2$

$3$

$4$

infinitely many

Answer: D

Concepts:completing the square Diophantine Equation bounding to limit cases

Difficulty rating: 1410

Solution:

Completing the square gives $x^{2020} + (y - 1)^2 = 1.$ Both terms are nonnegative, so $x^{2020} \le 1,$ forcing $x \in \{-1, 0, 1\}.$

If $x = 0,$ then $(y - 1)^2 = 1,$ giving $y = 0$ or $y = 2.$ If $x = \pm 1,$ then $x^{2020} = 1,$ so $(y - 1)^2 = 0$ and $y = 1.$ The solutions are $(0, 0), (0, 2), (1, 1),$ and $(-1, 1)$ — four in all.

Thus, the correct answer is D.

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