2023 AMC 12A Problem 10

Problem 10 of 25EasierAlgebra

Positive real numbers

x

and

y

satisfy

y^3=x^2

and

(y-x)^2=4y^2.

What is

x+y?

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Solution

From

(y-x)^2=4y^2

we get

y-x=\pm 2y.

The choice

y-x=2y

gives

x=-y\lt 0,

impossible, so

y-x=-2y,

meaning

x=3y.

Substituting into

y^3=x^2=9y^2

gives

y=9,

hence

x=27

and

x+y=36.

Thus, the correct answer is D.

Concepts: system of equations · substitution

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.