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Difference between revisions of "1999 AHSME Problems/Problem 10"

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==Problem==
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A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.
 
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.
  
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<math> \textbf{(A)}\  \text{I is true.} \qquad \textbf{(B)}\  \text{I is false.}\qquad \textbf{(C)}\ \text{II is true.} \qquad \textbf{(D)}\ \text{III is true.} \qquad \textbf{(E)}\ \text{IV is false.}</math>
 
<math> \textbf{(A)}\  \text{I is true.} \qquad \textbf{(B)}\  \text{I is false.}\qquad \textbf{(C)}\ \text{II is true.} \qquad \textbf{(D)}\ \text{III is true.} \qquad \textbf{(E)}\ \text{IV is false.}</math>
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==Solution==
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Three of the statements are correct, and only one digit is on the card. Thus, one of I and III are false. Therefore, II and IV must both be true. The answer is therefore <math>\boxed{\textbf{(C)}}</math>.
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==See Also==
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{{AHSME box|year=1999|num-b=9|num-a=11}}
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{{MAA Notice}}

Latest revision as of 13:34, 5 July 2013

Problem

A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.

I. The digit is 1. II. The digit is not 2. III. The digit is 3. IV. The digit is not 4.

Which one of the following must necessarily be correct?

$\textbf{(A)}\ \text{I is true.} \qquad \textbf{(B)}\ \text{I is false.}\qquad \textbf{(C)}\ \text{II is true.} \qquad \textbf{(D)}\ \text{III is true.} \qquad \textbf{(E)}\ \text{IV is false.}$

Solution

Three of the statements are correct, and only one digit is on the card. Thus, one of I and III are false. Therefore, II and IV must both be true. The answer is therefore $\boxed{\textbf{(C)}}$.

See Also

1999 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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