Maths Olympiad Prep

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Combinatorics Difficulty 5.9 AIME, harder Prove it

## Task 3 - 230623

The four students Erdbach, Freimuth, Giebler, and Hausmann have the first names Alfred, Bernd, Christian, and Detlef (possibly not in this order).

They met at Siegfried Zander's birthday party. The following is known:

(1) As the first guest, Siegfried could greet his classmate Hausmann, the second guest was Christian, and then Erdbach. Last to arrive was Bernd.

(2) Each of these four guests brought exactly one gift for the birthday person: Hausmann brought a dice game, Alfred a pen, Bernd a bouquet of roses, and Giebler a book.

Show that from these statements, the four birthday guests' first and last names can be uniquely determined!

Provide these matching first and last names!

Solution

Due to (1), Hausmann is neither Christian nor Bernd. Due to (2), he is also not Alfred. Therefore: (3) Hausmann has the first name Detlef.

Due to (2), Giebler is neither Alfred nor Bernd, and due to (3), he is also not Detlef. Therefore: (4) Giebler has the first name Christian.

Due to (1), Erdbach is neither Christian nor Bernd, and due to (3), he is also not Detlef. Therefore: (5) Erdbach has the first name Alfred.

Due to (3), (4), and (5), only the first name Bernd remains for Freimuth. The matching names are thus: Alfred Erdbach, Bernd Freimuth, Christian Giebler, and Detlef Hausmann.

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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.