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China mathematical olympiad 1995 problem6

Source: China mathematical olympiad 1995 problem6

September 15, 2013
vectorfunctionnumber theory unsolvednumber theory

Problem Statement

Let n(n>1)n(n>1) be an odd. We define xk=(x1(k),x2(k),⋯ ,xn(k))x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n) as follow: x0=(x1(0),x2(0),⋯ ,xn(0))=(1,0,⋯ ,0,1)x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1); xi(k)={0,emsp;xi(k−1)=xi+1(k−1),1,emsp;xi(k−1)≠xi+1(k−1), x^{(k)}_i =\begin{cases}0,   x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1,   x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} i=1,2,⋯ ,ni=1,2,\cdots ,n, where xn+1(k−1)=x1(k−1)x^{(k-1)}_{n+1}= x^{(k-1)}_1. Let mm be a positive integer satisfying x0=xmx_0=x_m. Prove that mm is divisible by nn.
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