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triple (x, y, z) replaced by (y+z- x, z+x- y,x +y -z), a+b+c=2013, 10 steps

Source: Dutch BxMO/EGMO TST 2013 p2

August 24, 2019
Triplepositive integerscombinatoricsnumber theorySum

Problem Statement

Consider a triple (a,b,c)(a, b, c) of pairwise distinct positive integers satisfying a+b+c=2013a + b + c = 2013. A step consists of replacing the triple (x,y,z)(x, y, z) by the triple (y+z−x,z+x−y,x+y−z)(y + z - x,z + x - y,x + y - z). Prove that, starting from the given triple (a,b,c)(a, b,c), after 1010 steps we obtain a triple containing at least one negative number.
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