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maximal and the minimal values of x_1 - 2 * x_2 + x_3

Source: Vietnam TST 1993 for the 34nd IMO, problem 3

June 25, 2005
inequalities unsolvedinequalities

Problem Statement

Let's consider the real numbers x1,x2,x3,x4x_1, x_2, x_3, x_4 satisfying the condition 12≤x12+x22+x32+x42≤1 \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 Find the maximal and the minimal values of expression: A=(x1−2⋅x2+x3)2+(x2−2⋅x3+x4)2+(x2−2⋅x1)2+(x3−2⋅x4)2 A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2
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