1984 AMC 12 Problem 28

Question

The number of distinct pairs of integers

(x,y)

such that

\begin{aligned} 0&\lt x\lt y,\\ \sqrt{1984}&=\sqrt{x}+\sqrt{y} \end{aligned}

is

0

1

2

3

7

Use

1984=31\cdot8^2

and write both radicands with a common squarefree part

Set

x=31u^2,\ y=31v^2

and count positive integers

u\lt v

with

u+v=8

Because

\sqrt{x}+\sqrt{y}=8\sqrt{31},

the two radicals must have common squarefree part

31.

Write

x=31u^2

and

y=31v^2

for positive integers

u,v.

Then

u+v=8,\qquad u\lt v.

The possibilities are

(u,v)=(1,7),

(2,6),

and

(3,5),

producing three distinct pairs

(x,y).

Therefore, the correct answer is D.

Problem 28 in other years