Problem

Source: Turkey TST 2001 - P3

Tags: modular arithmetic, parameterization, number theory proposed, number theory



For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$ $(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)