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5

Part of 2004 Moldova Team Selection Test

Problems(1)

f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{

Source: Moldova TST 2004

3/8/2023
Let n∈Nn\in\mathbb{N}, the set A={(x1,x2...,xn)∣xi∈R+,i=1,2,...,n}A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\} and the function f:A→R,f(x1,...,xn)=1x1+12x2+…+1(n−1)xn−1+1nxn.f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}. Prove that f((n1),(n2),...,(nn−1),(nn))=f(2n−1,2n−2,...,2,1).f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).
Binomial