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system of inequalities and equations

Source: Polish second round 2006

February 24, 2006
inequalitiessymmetryalgebra proposedalgebra

Problem Statement

Positive integers a,b,c,x,y,za,b,c,x,y,z satisfy: a2+b2=c2a^2+b^2=c^2, x2+y2=z2x^2+y^2=z^2 and ∣x−a∣≤1|x-a| \leq 1 , ∣y−b∣≤1|y-b| \leq 1. Prove that sets {a,b}\{a,b\} and {x,y}\{x,y\} are equal.
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