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Maxima with 6 inequalities and how many ways to obtain it.

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October 4, 2010
inequality systemalgebramaximizationIMO ShortlistIMO Longlist

Problem Statement

(NET3)(NET 3) Let x1,x2,x3,x4,x_1, x_2, x_3, x_4, and x5x_5 be positive integers satisfying x1+x2+x3+x4+x5=1000,x_1 +x_2 +x_3 +x_4 +x_5 = 1000, x1−x2+x3−x4+x5>0,x_1 -x_2 +x_3 -x_4 +x_5 > 0, x1+x2−x3+x4−x5>0,x_1 +x_2 -x_3 +x_4 -x_5 > 0, −x1+x2+x3−x4+x5>0,-x_1 +x_2 +x_3 -x_4 +x_5 > 0, x1−x2+x3+x4−x5>0,x_1 -x_2 +x_3 +x_4 -x_5 > 0, −x1+x2−x3+x4+x5>0-x_1 +x_2 -x_3 +x_4 +x_5 > 0 (a)(a) Find the maximum of (x1+x3)x2+x4(x_1 + x_3)^{x_2+x_4} (b)(b) In how many different ways can we choose x1,...,x5x_1, . . . , x_5 to obtain the desired maximum?
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