# 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.