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max S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1)

Source: Argentina 1997 OMA L3 p3

May 13, 2024
algebraSuminequalities

Problem Statement

Let x1,x2,x3,…,x100x_1,x_2,x_3,\ldots ,x_{100} be one hundred real numbers greater than or equal to 00 and less than or equal to 11. Find the maximum possible value of the sumS=x1(1−x2)+x2(1−x3)+x3(1−x4)+⋯+x99(1−x100)+x100(1−x1).S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).
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