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min of x_1 +x_2^2/ 2 + x_3^3/ 3 +...+ x_n^n /n if sum 1/x_i=n

Source: 2005 Moldova JBMO TST p6

February 20, 2021

algebrainequalities

Problem Statement

Let

n

be a nonzero natural number, and

x_1, x_2,..., x_n

positive real numbers that

\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n

. Find the minimum value of the expression

x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}