1982 AMC 12 Problem 20

Attempt Problem 20 of the 1982 AMC 12 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 1982 AMC 12 solutions, or check the answer key.

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20.

The number of pairs of positive integers

(x,y)

which satisfy the equation

x^2+y^2=x^3

0

1

2

not finite

none of these

Rearrange as

y^2=x^2(x-1)

Choose

x-1

to be a perfect square

For every positive integer

k,

take

x=k^2+1

and

y=k(k^2+1)=kx.

Then

y^2=k^2x^2=x^2(x-1),

x^2+y^2=x^3.

This supplies infinitely many positive-integer pairs.

Therefore, the correct answer is D.

Problem 20 in other years