A music talent show has 100 contestants and 10,000 viewers. Each viewer picks their top 5 performers. Any performer who is nominated by at least 20% of the viewers gets a prize. What is the maximum number of prizes that could be awarded?
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For a contestant to get a prize, they need at least 0.2⋅10000=2000 viewers nominating them. There are five batches of 2000 viewers (since there are 10000 total viewers), so let everyone in each batch nominate the same 5 people (this is 5 because that is the maximum number of contestants each viewer can nominate) and let the people nominated be different between batches. This gives us that the maximum number of contestants that can win prizes is 5⋅5=25 people.
So, at most 25 people can win a prize given the constraints.
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