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system 6x4 , x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4

Source: 2021 Dutch IMO TST 1.2

December 28, 2021

algebrasystem of equationsSystem

Problem Statement

Find all quadruplets

(x_1, x_2, x_3, x_4)

of real numbers such that the next six equalities apply:

\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}