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== | ||
+ | |||
+ | 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. | ||
+ | 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. | ||
+ | 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. | ||
+ | Using their responses, we get that <math>\boxed{C}</math> dropped the glass. | ||
==See Also== | ==See Also== |
Revision as of 18:56, 6 October 2018
Problem
During the rest hour, one of five students (,
,
,
, and
) dropped a glass of water. The following are the responses of the children when the teacher questioned them:
: It was
or
who dropped it.
: Neither
nor I did it.
: Both
and
are lying.
: Only one of
or
is telling the truth.
:
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 is telling the truth. Then
,
and one of
and
are lying. But only two students are lying, so there is a contradiction. Hence
is lying.
Then there are two possible scenarios: both
and
are lying, or both
and
are telling the truth.
Suppose that both
and
are lying. Then there are three students (
,
and
) who are lying, a contradiction. So both
and
are telling the truth.
Using their responses, we get that
dropped the glass.