2020 AMC 12B Problem 8

Below is the professionally curated solution for Problem 8 of the 2020 AMC 12B, 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.

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Concepts:completing the square Diophantine Equation bounding to limit cases

Difficulty rating: 1410

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

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

$1$

$2$

$3$

$4$

infinitely many

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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