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OlympiadBench / 1845 / Determine the largest real number a such that for all n≥1 and for…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
question
Determine the largest real number such that for all and for all real numbers satisfying , we have
$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\frac{n+1}{x_{n}}\right) .\tag{1}
$$Plain-text mathematical notation (without MathML)
Determine the largest real number 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
$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\frac{n+1}{x_{n}}\right) .\tag{1}
$$Original LaTeX notation
Determine the largest real number $a$ such that for all $n \geqslant 1$ and for all real numbers $x_{0}, x_{1}, \ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}$, we have
$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\frac{n+1}{x_{n}}\right) .\tag{1}
$$answer type
Numerical
is multiple answer
false
language
English
question type
Open-ended
subject
Math
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