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Maximum value of sum of 12th powers of x_i

Source: Chinese Maths Olympiad 1997 Problem 1

August 26, 2013

functioninequalitiesinequalities unsolved

Problem Statement

Let

x_1,x_2,\ldots ,x_{1997}

be real numbers satisfying the following conditions: i)

-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}

for

i=1,2,\ldots ,1997

; ii)

x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}

. Determine (with proof) the maximum value of

x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}