2001 AMC 12 Problem 7

Below is the professionally curated solution for Problem 7 of the 2001 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 12 solutions, or check the answer key.

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Concepts:Diophantine Equation prime factorization bounding to limit cases

Difficulty rating: 1370

A charity sells $140$ benefit tickets for a total of $\$2001.$ Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?

$\$782$

$\$986$

$\$1158$

$\$1219$

$\$1449$

Solution:

Let $n$ tickets sell at full price $p$ dollars. Then $np + (140 - n)\dfrac{p}{2} = 2001,$ so $p(n + 140) = 4002 = 2 \cdot 3 \cdot 23 \cdot 29.$

Since $0 \le n \le 140,$ we need a factor of $4002$ with $140 \le n + 140 \le 280.$ The only such factor is $174 = 2 \cdot 3 \cdot 29,$ giving $n = 34$ and $p = 23.$

The full-price tickets raise $34 \cdot 23 = 782$ dollars.

Thus, the correct answer is A.

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