Avoid This Common Mistake Made With Fractions

Algebra Level 3

Sleepy was sitting in class hearing his teacher talk about fractions. He learns that

xn+yn=(x+y)n because x+y=(x+y) ` \frac{x}{n} + \frac{y}{n} = \frac{ (x+y)} { n} \text{ because } x + y = (x+y) '

before falling asleep. As punishment for this misdemeanor, his teacher made him solve 1x+1y \frac{1}{x} + \frac{1}{y} on the board. Drawing from the previous example, he said that 1x+1y=1(x+y). \frac{ 1}{x} + \frac{1}{y} = \frac{1}{(x+y)}.

How many ordered pairs of integers subject to 100x100,100y100 -100 \leq x \leq 100, -100 \leq y \leq 100 are there, such that

1x+1y=1(x+y)? \frac{ 1}{x} + \frac{1}{y} = \frac{1}{(x+y)}?

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