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3

Part of 1975 Czech and Slovak Olympiad III A

Problems(1)

Cyclic system with 6 variables

Source: Czech and Slovak Olympiad 1975, National Round, Problem 3

7/20/2024
Determine all real tuples (x1,x2,x3,x4,x5,x6)\left(x_1,x_2,x_3,x_4,x_5,x_6\right) such that \begin{align*} x_1(x_6 + x_2) &= x_3 + x_5, \\ x_2(x_1 + x_3) &= x_4 + x_6, \\ x_3(x_2 + x_4) &= x_5 + x_1, \\ x_4(x_3 + x_5) &= x_6 + x_2, \\ x_5(x_4 + x_6) &= x_1 + x_3, \\ x_6(x_5 + x_1) &= x_2 + x_4. \end{align*}
algebraCyclicsystem of equationsReal Roots