Suppose that we run Gale and Shapley's algorithm for n men and n women where n is an even integer. In the solution that Gale and Shapley's algorithm finds, suppose that a man or a women is married to someone in the first half of his or her preference list, then we say he or she is happy with the matching. Prove that in the matching output by Gale and Shapley's algorithm, at least one person (man or woman) is happy with the matching.
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