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7

Part of 2010 Canadian Mathematical Olympiad Qualification Repechage

Problems(1)

Function g for triples of real numbers

Source: Canadian Repêchage 2010: Problem 7

5/6/2014
If (a, b, c)(a,~b,~c) is a triple of real numbers, de fine
[*] g(a, b, c)=(a+b, b+c, a+c)g(a,~b,~c)=(a+b,~b+c,~a+c), and [*] gn(a, b, c)=g(gn−1(a, b, c))g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c)) for n≥2n\ge 2
Suppose that there exists a positive integer nn so that gn(a, b, c)=(a, b, c)g^n(a,~b,~c)=(a,~b,~c) for some (a, b, c)≠(0, 0, 0)(a,~b,~c)\neq (0,~0,~0). Prove that g6(a, b, c)=(a, b, c)g^6(a,~b,~c)=(a,~b,~c)
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