Discrete Mathematics
# Pigeonhole Principle

There are \(22\) people in a party. Calvin, one of the participants of the party, shakes hands with \(18\) friends forgetting about the other three, goes to the washroom to wash his hands, and returns to the party. Then he again shakes hands with \(18\) friends, goes to the washroom, and returns to the party. He repeats this same pattern over and over.

Calvin sits down with some punch immediately after a final round of \(18\) handshakes in which Calvin shakes hands with one final friend who he's randomly managed to neglect all the rest of the evening. Sipping his punch, Calvin realizes that the numbers of handshakes with each of the \(21\) friends are all different. What is the minimum number of times he must have returned to the party, assuming that he always shakes hands with \(18\) friends after every return?

Alice, Bob, Candice and David run for school president. There are \(201\) students in the school and each student can vote for only one candidate. The person with the largest number of votes is the winner. If every student votes, what is the minimum number of votes needed for it to be at least possible to win the election?

**Details and assumptions**

Ties for the winner are not allowed in this election.

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