2023 AMC 12A Problem 10

Attempt Problem 10 of the 2023 AMC 12A 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 2023 AMC 12A solutions, or check the answer key.

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10.

Positive real numbers

x

and

y

satisfy

y^3=x^2

and

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

What is

x+y?

12

18

24

36

42

Taking square roots,

y-x=\pm 2y

Positivity rules out one sign and forces

x=3y;

substitute into

y^3=x^2

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.

Problem 10 in other years