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2001 AMC 10 Problem 14

Problem 14 of 25IntermediateNumber TheoryProblem-Solving Techniques

A charity sells 140140 benefit tickets for a total of $2001.\$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?

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Solution

Let nn full-price tickets sell at pp dollars each. Then np+(140−n)p2=2001,np+(140-n)\dfrac p2=2001, so p(n+140)p(n+140) =4002=4002 =2⋅3⋅23⋅29.=2\cdot3\cdot23\cdot29. Since 140≤n+140≤280,140\le n+140\le280, the only factor of 40024002 in range is 174=2⋅3⋅29.174=2\cdot3\cdot29. So n+140=174,n+140=174, giving n=34n=34 and p=23.p=23. The full-price tickets raise 34⋅23=78234\cdot23=782 dollars. Thus, the correct answer is A.
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