Olympiad problems

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Found problems: 2

2023 Belarusian National Olympiad, 10.8

On the Alphamegacentavra planet there are $2023$ cities, some of which are connected by non-directed flights. It turned out that among any $4$ cities one can find two with no flight between them. Find the maximum number of triples of cities such that between any two of them there is a flight.

2023 Euler Olympiad, Round 2, 6

Tags: combinatorics , turan , euler

Let $n$ be some positive integer. Free university accepts $n^2$ freshmen, where no two students know each other initially. It's known that students can only get to know eachother on parties, which are organized by the university's administration. The administration's goal is to ensure that there does not exist a group of $n$ students where none of them know each other. Organizing a party with $m$ members incurs a cost of $m^2 - m$. Determine the minimal cost for the administration to fulfill their goal. [i]Proposed by Luka Macharashvili, Georgia[/i]