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Omni-MATH / Let n>k be two natural numbers and let a₁,…,a_(n) be real numbers in the open interval (k−1,k). Let x₁,…,x_(n) be positive reals such that for any subset I⊂{1,…,n} satisfying |I|=k…

Problem

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problem

Let n>kn > k be two natural numbers and let a1,…,ana_1,\ldots,a_n be real numbers in the open interval (k−1,k)(k-1,k). Let x1,…,xnx_1,\ldots,x_n be positive reals such that for any subset I⊂{1,…,n}I \subset \{1,\ldots,n \} satisfying |I|=k|I| = k, one has ∑i∈Ixi≤∑i∈Iai.\sum_{i \in I} x_i \leq \sum_{i \in I} a_i. Find the largest possible value of x1x2⋯xnx_1 x_2 \cdots x_n.
Plain-text mathematical notation (without MathML)
Let n>k be two natural numbers and let a₁,…,a_(n) be real numbers in the open interval (k−1,k). Let x₁,…,x_(n) be positive reals such that for any subset I⊂{1,…,n} satisfying |I|=k, one has 
∑_(i∈I)x_(i)≤∑_(i∈I)a_(i).
Find the largest possible value of x₁x₂⋯x_(n).
Original LaTeX notation
Let $n > k$ be two natural numbers and let $a_1,\ldots,a_n$ be real numbers in the open interval $(k-1,k)$. Let $x_1,\ldots,x_n$ be positive reals such that for any subset $I \subset \{1,\ldots,n \}$ satisfying $|I| = k$, one has 
$$\sum_{i \in I} x_i \leq \sum_{i \in I} a_i.$$
Find the largest possible value of $x_1 x_2 \cdots x_n$.

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Source and history

task_id
14ac5213-0750-54e2-ab8d-57df286c7eeb
task_key
test--14ac5213-0750-54e2-ab8d-57df286c7eeb
task_revision_id
1
config / split
/ test

Official source

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