2020 AMC 10B Problem 9

Attempt Problem 9 of the 2020 AMC 10B 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 10B solutions, or check the answer key.

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9.

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:Diophantine Equation completing the square bounding to limit cases

Difficulty rating: 1070

Small hint:

Complete the square in

y

Big hint:

The equation forces

x^{2020}\le 1,

so only a few integer

x

-values are possible

Solution:

Move all terms to one side and complete the square:

\begin{aligned} &x^{2020}+y^2=2y \\ &\quad \Longrightarrow x^{2020}+(y-1)^2=1. \end{aligned}

Because

(y-1)^2\ge 0,

we must have

x^{2020}\le 1.

Since

x

is an integer,

x=-1,0,1.

If

x=\pm1,

then

(y-1)^2=0,

so

y=1.

If

x=0,

then

(y-1)^2=1,

so

y=0

or

2.

This gives

4

ordered pairs.

Thus, D is the correct answer.