1969 AHSME Problems/Problem 19

Problem

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:

$\text{(A) } 0\quad \text{(B) } 3\quad \text{(C) } 4\quad \text{(D) } 12\quad \text{(E) } \infty$

Solution

Let $(xy)^2=a$ . The expression given is equal to $a^2-10a+9=0$ , which can be factored as $(a-9)(a-1)=0$ . Because $x$ and $y$ are positive, we can eliminate the possibilities where $a$ is negative. Thus, we have $a=(xy)^2=9$ and $a=(xy)^2=1$ . From here it is easy to see that the only integral pairs of $x$ and $y$ are $(3, 1), (1, 3)$ , and $(1, 1)$ . The answer is $\fbox{B}$ .

1969 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
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1969 AHSME Problems/Problem 19

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