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Omni-MATH / Find the largest real constant a such that for all n≥1 and for all real numbers x₀,x₁,...,x_(n) satisfying 0=x₀<x₁<x₂<⋯<x_(n) we have …

Problem

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problem

Find the largest real constant aa such that for all n≥1n \geq 1 and for all real numbers x0,x1,...,xnx_0, x_1, ... , x_n satisfying 0=x0<x1<x2<⋯<xn0 = x_0 < x_1 < x_2 < \cdots < x_n we have 1x1−x0+1x2−x1+…+1xn−xn−1≥a(2x1+3x2+…+n+1xn)\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)
Plain-text mathematical notation (without MathML)
Find the largest real constant a such that for all n≥1 and for all real numbers x₀,x₁,...,x_(n) satisfying 0=x₀<x₁<x₂<⋯<x_(n) we have
(1)/(x₁−x₀)+(1)/(x₂−x₁)+…+(1)/(x_(n)−x_(n−1))≥a((2)/(x₁)+(3)/(x₂)+…+(n+1)/(x_(n)))
Original LaTeX notation
Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]

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Source and history

task_id
b6b57a65-cee4-5bb3-8084-24350901adbf
task_key
test--b6b57a65-cee4-5bb3-8084-24350901adbf
task_revision_id
1
config / split
/ test

Official source

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