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Number theory Difficulty 4.0 AIME Find the answer United Kingdom

Find all (positive or negative) integers $n$ for which $n^2 + 20n + 11$ is a perfect square. Remember that you must justify that you have found them all.

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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.