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Difference between revisions of "2000 PMWC Problems/Problem I12"

(Created page with "==Problem== During the rest hour, one of five students (<math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math>) dropped a glass of water. The followin...")
 
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==Solution==
 
==Solution==
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Suppose that <math>D</math> is telling the truth. Then <math>C</math>, <math>E</math> and one of <math>A</math> and <math>B</math> are lying. But only two students are lying, so there is a contradiction. Hence <math>D</math> is lying.
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Then there are two possible scenarios: both <math>A</math> and <math>B</math> are lying, or both <math>A</math> and <math>B</math> are telling the truth.
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Suppose that both <math>A</math> and <math>B</math> are lying. Then there are three students (<math>A</math>, <math>B</math> and <math>D</math>) who are lying, a contradiction. So both <math>A</math> and <math>B</math> are telling the truth.
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Using their responses, we get that <math>\boxed{C}</math> dropped the glass.
  
 
==See Also==
 
==See Also==
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Back to test: https://artofproblemsolving.com/wiki/index.php/2000_PMWC_Problems

Latest revision as of 11:04, 23 December 2019

Problem

During the rest hour, one of five students ($A$, $B$, $C$, $D$, and $E$) dropped a glass of water. The following are the responses of the children when the teacher questioned them:

  • $A$: It was $B$ or $C$ who dropped it.
  • $B$: Neither $E$ nor I did it.
  • $C$: Both $A$ and $B$ are lying.
  • $D$: Only one of $A$ or $B$ is telling the truth.
  • $E$: $D$ is not speaking the truth.

The class teacher knows that three of them NEVER lie while the other two ALWAYS lie. Who dropped the glass?

Solution

Suppose that $D$ is telling the truth. Then $C$, $E$ and one of $A$ and $B$ are lying. But only two students are lying, so there is a contradiction. Hence $D$ is lying. Then there are two possible scenarios: both $A$ and $B$ are lying, or both $A$ and $B$ are telling the truth. Suppose that both $A$ and $B$ are lying. Then there are three students ($A$, $B$ and $D$) who are lying, a contradiction. So both $A$ and $B$ are telling the truth. Using their responses, we get that $\boxed{C}$ dropped the glass.

See Also

Back to test: https://artofproblemsolving.com/wiki/index.php/2000_PMWC_Problems

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