Maths Olympiad Prep

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Combinatorics Difficulty 4.8 AIME Find the answer Italy

Problem:

At a table, there are four people: Luca, Maria, Nicola and Paola. Each of the four either always lies or never lies. Moreover they do not like to talk about themselves, but rather about their friends; so much so that when asked who among them always lies, their answers are:

Luca: "every girl is always truthful"
Maria: "every boy is always a liar"
Nicola: "there is one girl who always lies, the other is always truthful"
Paola: "one of the boys is always truthful, the other never".

Can you say how many at the table are always truthful?

Pick one

Solution

Solution:

The answer is (C)\mathbf{(C)}. Certainly Luca cannot be truthful, since the two girls contradict each other, so they cannot both be telling the truth. If Maria were telling the truth, Nicola would be lying; therefore, it would not be true that there is a truthful girl, and in particular Maria would not be one, contradiction. Hence, Maria lies. So there is at least one boy who always tells the truth, and it must necessarily be Nicola. Therefore, by what he says, one of the two girls is truthful, and since it cannot be Maria, it is Paola. On the other hand, the latter states that there is only one boy who tells the truth, and from what was deduced earlier, this statement is true. Hence, the truthful ones are Nicola and Paola.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.