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Omni-MATH / Let (a_(n)) be the sequence of reals defined by a₁=(1)/(4) and the recurrence a_(n)=(1)/(4)(1+a_(n−1))²,n≥2. Find the minimum real λ such that for any non-negative reals …

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problem

Let (an) \left(a_{n}\right) be the sequence of reals defined by a1=14 a_{1}=\frac{1}{4} and the recurrence an=14(1+an−1)2,n≥2 a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2. Find the minimum real λ \lambda such that for any non-negative reals x1,x2,…,x2002 x_{1},x_{2},\dots,x_{2002}, it holds ∑k=12002Ak≤λa2002, \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, where Ak=xk−k(xk+⋯+x2002+k(k−1)2+1)2,k≥1 A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1.
Plain-text mathematical notation (without MathML)
Let (a_(n)) be the sequence of reals defined by a₁=(1)/(4) and the recurrence a_(n)=(1)/(4)(1+a_(n−1))²,n≥2. Find the minimum real λ such that for any non-negative reals x₁,x₂,…,x₂₀₀₂, it holds
∑_(k=1)²⁰⁰²A_(k)≤λa₂₀₀₂,
where A_(k)=(x_(k)−k)/((x_(k)+⋯+x₂₀₀₂+(k(k−1))/(2)+1)²),k≥1.
Original LaTeX notation
Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds
\[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \]
where $  A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

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Source and history

task_id
44a5da69-a628-5e31-b2e2-1b03b9aaa1ea
task_key
test--44a5da69-a628-5e31-b2e2-1b03b9aaa1ea
task_revision_id
1
config / split
/ test

Official source

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