6. (8 points) At the University of Innopolis, $2 n$ applicants who were not acquainted with each other were admitted to the first year. How many pairs of freshmen need to be introduced to each other to ensure that there will definitely be a trio of mutually acquainted individuals?
Solution: The problem is solved by a simple application of Turán's theorem for $K_{3}$: the maximum number of edges in a graph without triangles is $n^{2}$. An example of such a graph is the complete bipartite graph $K_{n, n}$. In the end, the answer to the problem will be the number $n^{2}+1$.

Answers:

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| Answer | 2602 | 2705 | 2810 | 2917 | 3026 | 3137 | 3250 | 3365 | 3482 | 2501 | 3601 |