2000 PMWC Problems/Problem I12

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.

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2000 PMWC Problems/Problem I12

Problem

Solution

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