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x_1x_2...x_n >= (n -1)^n if sum 1/(1+x_k)=1

Source: 2011 Saudi Arabia Pre-TST February 4.3

January 1, 2022
algebrainequalities

Problem Statement

Let x1,x2,...,xnx_1,x_2,...,x_n be positive real numbers for which 11+x1+11+x2+...+11+xn=1\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1 Prove that x1x2...xn≥(n−1)nx_1x_2...x_n \ge (n -1)^n.
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