Maths Olympiad Prep

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Combinatorics Difficulty 6.0 National olympiad Prove it

## 75. The Case of the McGregor Shop Robbery...

Mr. McGregor, the owner of a shop in London, reported to Scotland Yard by telephone that he had been robbed. Three repeat offenders, A, B, and C, suspected of the robbery, were summoned for interrogation. The following facts were established:

1) Each of the three suspects A, B, and C visited the shop on the day of the robbery, and no one else entered the shop that day.
2) If A is guilty, then he had exactly one accomplice.
3) If B is not guilty, then C is also not guilty.
4) If exactly two suspects are guilty, then A is one of them.
5) If C is not guilty, then B is also not guilty.

Against whom did Inspector Craig bring charges?

Solution

75. Inspector Craig brought charges against Mr. McGregor for attempting to mislead the police: there was no robbery in reality. Here is how Inspector Craig reasoned.

First step. Suppose A was guilty. Then, by statement (2), he would have had exactly one accomplice - no more, no less. Therefore, one of B, C would be guilty, and the other not guilty. But this contradicts statements (3) and (5), from which, taken together, it follows that B and C are either both guilty or both not guilty. Therefore, A must be not guilty.

Second step. From statements (3) and (5) it follows that B and C are either both guilty or both not guilty. If they were both guilty, there would be no other guilty parties (since A is not guilty). Therefore, in this case, there would be exactly two guilty parties. By statement (4), this would mean that A is guilty. Thus, we would arrive at a contradiction, since A is not guilty. Therefore, B and C are both not guilty.

Third step. Thus, it is established that A, B, and C are not guilty. But, as follows from statement (1), no one other than A, B, and C entered the shop on the day of the robbery and could not have committed it. Therefore, there was no robbery, and McGregor was lying.

Epilogue. Unable to withstand the irrefutable arguments of Inspector Craig, McGregor confessed that he had lied in the hope of obtaining insurance.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.