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Algebra Difficulty 6.2 National olympiad Prove it

Theorem 5 Let x1,x2,⋯ ,xn∈R+x_{1}, x_{2}, \cdots, x_{n} \in R^{+}, then
(∑i=1nxi)⋅(∑i=1nxi−1)≥n2\left(\sum_{i=1}^{n} x_{i}\right) \cdot\left(\sum_{i=1}^{n} x_{i}^{-1}\right) \geq n^{2}

Equality holds if and only if x1=x2=⋯=xnx_{1}=x_{2}=\cdots=x_{n}.

Solution

Proof: According to the above theorem, take α=1,δ=−1,γ=\alpha=1, \delta=-1, \gamma= β=0\beta=0. This completes the proof, or it can be directly proven by Corollary 4.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.