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Given a sequence, find all values of this expression

Source: Czech-Polish-Slovak 2002 Q1

April 28, 2013
inequalitiesceiling functionfloor functionalgebra unsolvedalgebra

Problem Statement

Let a,ba, b be distinct real numbers and k,mk,m be positive integers k+m=n≥3,k≤2m,m≤2kk + m = n \ge 3, k \le 2m, m \le 2k. Consider sequences x1,…,xnx_1,\dots , x_n with the following properties: (i) kk terms xix_i, including x1x_1, are equal to aa; (ii) mm terms xix_i, including xnx_n, are equal to bb; (iii) no three consecutive terms are equal. Find all possible values of xnx1x2+x1x2x3+⋯+xn−1xnx1x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1.
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