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MathNet 2016-part-01 — 184 problems (EN–VI) Question 73

Math Olympiad Vault · MathNet 2016-part-01 — 184 problems (EN–VI)

Question 73

There are $n$ students at a school. It is known that, for any two students $A$ and $B$, either $A$ loves $B$ or $B$ loves $A$ (but not both). (A student may thus be in love with several other students.) A *love triangle* is a configuration of three students $A$, $B$, $C$ such that $A$ loves $B$, $B$ loves $C$ and $C$ loves $A$. What is the maximal number of love triangles, given the number $n$ of students?