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Minimum of sum of n variables under constraint

Source: Romanian TST for 2019 IMO

October 1, 2019
inequalitiesminimumalgebra

Problem Statement

Let be a natural number n≥3. n\ge 3. Find inf⁡1=P(x1,x2,…,xn)x1,x2,…,xn∈R>0∑i=1n(1xi−xi), \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , where P(x1,x2,…,xn):=∑i=1n1xi+n−1, P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , and find in which circumstances this infimum is attained.
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