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Omni-MATH / Let n>2 be an integer and let ℓ∈{1,2,…,n}. A collection A₁,…,A_(k) of (not necessarily distinct) subsets of {1,2,…,n} is called ℓ-large if |A_(i)|≥ℓ for all 1≤i≤k. Find, in terms o…

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problem

Let n>2n > 2 be an integer and let ℓ∈{1,2,…,n}\ell \in \{1, 2,\dots, n\}. A collection A1,…,AkA_1,\dots,A_k of (not necessarily distinct) subsets of {1,2,…,n}\{1, 2,\dots, n\} is called ℓ\ell-large if |Ai|≥ℓ|A_i| \ge \ell for all 1≤i≤k1 \le i \le k. Find, in terms of nn and ℓ\ell, the largest real number cc such that the inequality ∑i=1k∑j=1kxixj|Ai∩Aj|2|Ai|⋅|Aj|≥c(∑i=1kxi)2 \sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2 holds for all positive integer kk, all nonnegative real numbers x1,x2,…,xkx_1,x_2,\dots,x_k, and all ℓ\ell-large collections A1,A2,…,AkA_1,A_2,\dots,A_k of subsets of {1,2,…,n}\{1,2,\dots,n\}.
Plain-text mathematical notation (without MathML)
Let n>2 be an integer and let ℓ∈{1,2,…,n}. A collection A₁,…,A_(k) of (not necessarily distinct) subsets of {1,2,…,n} is called ℓ-large if |A_(i)|≥ℓ for all 1≤i≤k. Find, in terms of n and ℓ, the largest real number c such that the inequality
∑_(i=1)^(k)∑_(j=1)^(k)x_(i)x_(j)(|A_(i)∩A_(j)|²)/(|A_(i)|⋅|A_(j)|)≥c(∑_(i=1)^(k)x_(i))²
holds for all positive integer k, all nonnegative real numbers x₁,x₂,…,x_(k), and all ℓ-large collections A₁,A₂,…,A_(k) of subsets of {1,2,…,n}.
Original LaTeX notation
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$. A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$-large if $|A_i| \ge \ell$ for all $1 \le i \le k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality
\[  \sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2 \]
holds for all positive integer $k$, all nonnegative real numbers $x_1,x_2,\dots,x_k$, and all $\ell$-large collections $A_1,A_2,\dots,A_k$ of subsets of $\{1,2,\dots,n\}$.

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Source and history

task_id
f8642477-f53a-57e6-bac0-f2000b7305e5
task_key
test--f8642477-f53a-57e6-bac0-f2000b7305e5
task_revision_id
1
config / split
/ test

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