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Omni-MATH / Given positive integers n,k such that n≥4k, find the minimal value λ=λ(n,k) such that for any positive reals a₁,a₂,…,a_(n), we have …
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problem
Given positive integers such that , find the minimal value such that for any positive reals , we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where Plain-text mathematical notation (without MathML)
Given positive integers n,k such that n≥4k, find the minimal value λ=λ(n,k) such that for any positive reals a₁,a₂,…,a_(n), we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where a_(n+i)=a_(i),i=1,2,…,kOriginal LaTeX notation
Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where $a_{n+i}=a_i,i=1,2,\ldots,k$Discussion
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