2021 AMC 10A Problem 19

Problem 19 of 25HarderAlgebra Geometry

The area of the region bounded by the graph of

x^2+y^2 = 3|x-y| + 3|x+y|

m+n\pi,

where

m

and

n

are integers. What is

m + n?

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Solution

Consider the four sign cases for

x-y

and

x+y.

In one case, for example,

|x-y|=x-y

and

|x+y|=x+y,

\begin{gathered} x^2+y^2=6x \\ \quad\Longrightarrow\quad (x-3)^2+y^2=9. \end{gathered}

The other three cases similarly give circles of radius

3

centered at

(0,3),

(-3,0),

and

(0,-3).

The relevant arcs form the boundary shown by these four congruent circle pieces. The region consists of a central square of side length

6,

together with four semicircles of radius

3.

The square contributes area

36,

and the four semicircles have the area of two full radius-

3

circles, namely

18\pi.

Therefore the area is

36+18\pi,

m+n=36+18=54.

Thus, E is the correct answer.