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Minimum of $F_n$

Source: Bosnia and Herzegovina TST 2013 problem 5

May 20, 2013
inequalities proposedinequalities

Problem Statement

Let x1,x2,…,xnx_1,x_2,\ldots,x_n be nonnegative real numbers of sum equal to 11. Let Fn=x12+x22+⋯+xn2−2(x1x2+x2x3+⋯+xnx1)F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1). Find: a) min⁡F3\min F_3; b) min⁡F4\min F_4; c) min⁡F5\min F_5.
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