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Difference between revisions of "1978 AHSME Problems/Problem 22"

(solution)
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The following four statements, and only these are found on a card:
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<asy>
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pair A,B,C,D,E,F,G;
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A=(0,1);
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B=(0,5);
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C=(11,5);
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D=(11,1);
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E=(0,4);
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F=(0,3);
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G=(0,2);
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draw(A--B--C--D--cycle);
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label("On this card exactly one statement is false.", B, SE);
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label("On this card exactly two statements are false.", E, SE);
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label("On this card exactly three statements are false.", F, SE);
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label("On this card exactly four statements are false.", G, SE);
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</asy>
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(Assume each statement is either true or false.) Among them the number of false statements is exactly
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<math>\textbf{(A)}\ 0 \qquad
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\textbf{(B)}\ 1 \qquad
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\textbf{(C)}\ 2 \qquad
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\textbf{(D)}\ 3 \qquad
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\textbf{(E)}\ 4    </math>
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== Solution ==
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There can be at most one true statement on the card, eliminating <math>\textbf{(A)}, \textbf{(B)},</math> and <math>\textbf{(C)}</math>. If there are <math>0</math> true on the card, statement <math>4</math> ("On this card exactly four statements are false") will be correct, causing a contradiction. Therefore, the answer is <math>\textbf{(D)}\ 3</math>, since <math>3</math> are false and only the third statement ("On this card exactly three statements are false") is correct.
 
There can be at most one true statement on the card, eliminating <math>\textbf{(A)}, \textbf{(B)},</math> and <math>\textbf{(C)}</math>. If there are <math>0</math> true on the card, statement <math>4</math> ("On this card exactly four statements are false") will be correct, causing a contradiction. Therefore, the answer is <math>\textbf{(D)}\ 3</math>, since <math>3</math> are false and only the third statement ("On this card exactly three statements are false") is correct.

Revision as of 20:10, 29 December 2019

The following four statements, and only these are found on a card: [asy] pair A,B,C,D,E,F,G; A=(0,1); B=(0,5); C=(11,5); D=(11,1); E=(0,4); F=(0,3); G=(0,2); draw(A--B--C--D--cycle); label("On this card exactly one statement is false.", B, SE); label("On this card exactly two statements are false.", E, SE); label("On this card exactly three statements are false.", F, SE); label("On this card exactly four statements are false.", G, SE); [/asy]

(Assume each statement is either true or false.) Among them the number of false statements is exactly

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

There can be at most one true statement on the card, eliminating $\textbf{(A)}, \textbf{(B)},$ and $\textbf{(C)}$. If there are $0$ true on the card, statement $4$ ("On this card exactly four statements are false") will be correct, causing a contradiction. Therefore, the answer is $\textbf{(D)}\ 3$, since $3$ are false and only the third statement ("On this card exactly three statements are false") is correct.

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