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3

Part of 1989 China National Olympiad

Problems(1)

China Mathematical Olympiad 1989 problem3

Source: China Mathematical Olympiad 1989 problem3

10/30/2013
Let SS be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to 11). We define function f:S→Sf:S\rightarrow S as follow: ∀z∈S\forall z\in S, f(1)(z)=f(z),f(2)(z)=f(f(z)),…, f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots, f(k)(z)=f(f(k−1)(z))(k>1,k∈N),…f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots We call cc an nn-period-point of ff if cc (c∈Sc\in S) and nn (n∈Nn\in\mathbb{N}) satisfy: f(1)(c)≠c,f(2)(c)≠c,f(3)(c)≠c,…,f(n−1)(c)≠c,f(n)(c)=cf^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c. Suppose that f(z)=zmf(z)=z^m (z∈S;m>1,m∈Nz\in S; m>1, m\in \mathbb{N}), find the number of 19891989-period-point of ff.
functioncomplex numbersalgebra unsolvedalgebra