Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Find the answer

When will A say yes if A will say yes when B says no to $n-1$ or $n$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

A will say yes when B says no to $n-1$ or $n$ , as A will then know B's number is one greater than A's number. Thus, A responds first, after $\frac{n-1}{2}$ 'no' responses if $n$ is odd, after $\frac{n}{2}$ 'no' responses if $n$ is even.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.