Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Find the answer Italy

Problem:

On the Island of Knights (who always tell the truth) and Knaves (who always lie), a survey is conducted among the 2013 inhabitants, in which there are three questions: "Do you support team A?", "Do you support team B?" and "Do you support team C?". We know that every islander answers all three questions and supports one and only one of the three teams. If the "Yes" answers total 3000, how many of the islanders are Knights?

Pick one

Solution

Solution:

The answer is (C). Let $C$ be the number of knights and $F$ the number of knaves. Clearly, every knight interviewed will answer exactly one "Yes" (to the question relating to the team he supports) and two "No", while a knave will answer exactly one "No" and two "Yes".

The information provided by the problem is the total number of inhabitants of the island, $C+F$ , and the total number of Yes answers, that is, by what we just observed, $C+2F$ . We then have the linear system
$\left\{ \begin{array}{l} C+F=2013 \\ C+2F=3000 \end{array} \right.$
and subtracting the first equation from the second one gets $F=987$ , from which $C=2013-987=1026$ .

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