Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Find the answer

Problem 2. Misha, Anton, Petya, and Fyodor competed in parallel bars exercises. When asked who took which place, they answered:

Misha: - I was neither first nor last.

Anton: - I was not last.

Petya: - I was first.

Fyodor: - I was last.

It is known that one of the boys lied, while the other three told the truth. Which of the boys took the last place?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Answer: Fedia took the last place.

Solution. Suppose Fedia lied. Then no one of the boys could take the last place, which is impossible. Therefore, Fedia told the truth and was the last.

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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.