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Number theory Difficulty 3.0 AMC 10/12 Find the answer United Kingdom

Find all solutions in positive integers $x$ , $y$ , $z$ to the simultaneous equations
$x + y - z = 12$
$x^2 + y^2 - z^2 = 12.$

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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.