4

Part of 1966 German National Olympiad

Problems(1)

4x4 system x_1x2 + x_1x_3 + x_2x_3 + x_4 = 2

Source: 1965-66 Germany R4 12.4

Determine all ordered quadruples of real numbers

(x_1, x_2, x_3, x_4)

for which the following system of equations exists, is fulfilled:

x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2

x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2

x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2

x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2

algebrasystem of equationsSystem