Cyclic system with 6 variables
Source: Czech and Slovak Olympiad 1975, National Round, Problem 3
July 20, 2024
algebraCyclicsystem of equationsReal Roots
Problem Statement
Determine all real tuples 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*}