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Omni-MATH / For each integer n≥2, determine the largest real constant C_(n) such that for all positive real numbers a₁,…,a_(n) we have (a₁²+…+a_(n)²)/(n)≥((a₁+…+a_(n))/(n))²+C_(n)⋅(a₁−a_(n))².

Problem

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problem

For each integer n≥2n\geqslant2, determine the largest real constant CnC_n such that for all positive real numbers a1,…,ana_1, \ldots, a_n we have a12+…+an2n≥(a1+…+ann)2+Cn⋅(a1−an)2.\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}
Plain-text mathematical notation (without MathML)
For each integer n≥2, determine the largest real constant C_(n) such that for all positive real numbers a₁,…,a_(n) we have
(a₁²+…+a_(n)²)/(n)≥((a₁+…+a_(n))/(n))²+C_(n)⋅(a₁−a_(n))².
Original LaTeX notation
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]

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Source and history

task_id
5cceca62-70a7-55dc-b4a4-32d3b5a62677
task_key
test--5cceca62-70a7-55dc-b4a4-32d3b5a62677
task_revision_id
1
config / split
/ test

Official source

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