Brilliant's Mathematical Contest!

Logic Level 5

In a Mathematics Contest conducted by Brilliant, only the following facts are known to its staff:

  • The number of problems asked in the contest was \({n \geq 4}\).

  • Each problem was solved by exactly four contestants.

  • For each pair of problems, there is exactly one contestant who solved both the problems.

Assuming that the number of contestants is greater than or equal to \({4n}\), help the staff find the minimum value of \({n}\) for which there always exists a contestant who solved all the problems.

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