Consider a convention with one hundred politicians. Of them, at least one is honest and at least one is crooked. Of any random sample of two of them, at least one is crooked. How many of them are honest, and how many crooked?
The answer is that one is honest and the remainder are all crooked. Any higher number of honest politicians at the convention would permit a random pair without a single crooked one. Any lower number would be ruled out by the criteria stated in the problem.
The key to internalizing this is to convert the statement, "Of any random sample of two of them, at least one is crooked." to "It's impossible to pick two people who are both honest". If the latter is the case, then there can't be more than one honest person.
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