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Number theory Difficulty 5.0 AIME, harder Find the answer United Kingdom

Find all positive integers $n$ such that $n \times 2^n + 1$ is a square.

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.