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Hundred self-referential at-most statements

Question

A page lists 100100 claims. Claim number nn (for n=1,…,100n=1,\dots,100) reads: "No more than n−1n-1 of the claims on this page are true." How many of the claims are true?

Short answer

5050. If exactly TT claims are true, claim nn is true precisely when T≤n−1T\le n-1, which holds for n=T+1,…,100n=T+1,\dots,100, i.e. 100−T100-T claims. Consistency needs 100−T=T100-T=T, so T=50T=50.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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