Question 6
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$.
A collection $A_1$, …, $A_k$ 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 $x_1$, $x_2$, …, $x_k$,
and all $\ell$-large collections $A_1$, $A_2$, …, $A_k$
of subsets of $\{1,2,\dots,n\}$.
Proposed by Titu Andreescu, Gabriel Dospinescu. One nonofficial editorial point; manual essay grading.
A collection $A_1$, …, $A_k$ 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 $x_1$, $x_2$, …, $x_k$,
and all $\ell$-large collections $A_1$, $A_2$, …, $A_k$
of subsets of $\{1,2,\dots,n\}$.
Proposed by Titu Andreescu, Gabriel Dospinescu. One nonofficial editorial point; manual essay grading.