1984 AMC 12 Problem 1

Attempt Problem 1 of the 1984 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 1984 AMC 12 solutions, or check the answer key.

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\frac{1000^2}{252^2-248^2}

equals

62{,}500

1000

500

250

\frac12

Factor the denominator as a difference of squares

Use

252+248=500

and

252-248=4

Factoring the denominator gives

\begin{aligned} 252^2-248^2 &=(252+248)\\ &\quad{}\cdot(252-248)\\ &=500\cdot4=2000. \end{aligned}

Thus the given quotient is

\frac{1000^2}{2000}=500.

Therefore, the correct answer is C.

Problem 1 in other years