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Sistem of real numbers with sum of reciprocals

Source: Romanian IMO Team Selection Test TST 1988, problem 6

October 1, 2005
vectoralgebra proposedalgebra

Problem Statement

Find all vectors of nn real numbers (x1,x2,…,xn)(x_1,x_2,\ldots,x_n) such that {x1=1x2+1x3+⋯+1xnx2=1x1+1x3+⋯+1xn ⋯ xn=1x1+1x2+⋯+1xn−1 \left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right. Mircea Becheanu
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