1969 AMC 12 Problem 19

Attempt Problem 19 of the 1969 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 1969 AMC 12 solutions, or check the answer key.

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19.

The number of distinct ordered pairs

(x,y),

where

x

and

y

have positive integral values satisfying the equation

x^4y^4-10x^2y^2+9=0,

is:

0

3

4

12

infinite

Treat

x^2y^2

as one variable

After factoring, translate the possible values into conditions on the positive integer product

xy

Let

u=x^2y^2.

Then

\begin{aligned} u^2-10u+9&=0,\\ (u-1)(u-9)&=0. \end{aligned}

Since

x

and

y

are positive integers, this means

xy=1

xy=3.

The pairs are

(1,1),

(1,3),

and

(3,1),

for a total of

3.

Therefore, the correct answer is B.

Problem 19 in other years