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x_1+ x_2+ x_3 +x_4=1,a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c, ...

Source: Austrian Polish 1983 APMC

April 30, 2020
parametersystem of equationsalgebra

Problem Statement

Let a1<a2<a3<a4a_1 < a_2 < a_3 < a_4 be given positive numbers. Find all real values of parameter cc for which the system {x1+x2+x3+x4=1a1x1+a2x2+a3x3+a4x4=ca12x1+a22x2+a32x3+a42x4=c2\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\ a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases} has a solution in nonnegative (x1,x2,x3,x4)(x_1,x_2,x_3,x_4) real numbers.
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