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Show that the fraction isn't an integer - ILL 1990 COL2

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September 18, 2010
functionalgebrafunctional equationtriangle inequalityIMO ShortlistIMO Longlist

Problem Statement

Let function f:Z≥00→Nf : \mathbb Z_{\geq 0}^0 \to \mathbb N satisfy the following conditions:
(i) f(0,0,0)=1; f(0, 0, 0) = 1;
(ii) f(x,y,z)=f(x−1,y,z)+f(x,y−1,z)+f(x,y,z−1);f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);
(iii) when applying above relation iteratively, if any of x′,y′,zx', y', z' is negative, then f(x′,y′,z′)=0.f(x', y', z') = 0.
Prove that if x,y,zx, y, z are the side lengths of a triangle, then (f(x,y,z))kf(mx,my,mz)\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)} is not an integer for any integers k,m>1.k, m > 1.
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