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Difference between revisions of "2024 USAMO Problems/Problem 6"

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Let <math>n>2</math> be an integer and let <math>\ell \in\{1,2, \ldots, n\}</math>. A collection <math>A_1, \ldots, A_k</math> of (not necessarily distinct) subsets of <math>\{1,2, \ldots, n\}</math> is called <math>\ell</math>-large if <math>\left|A_i\right| \geq \ell</math> for all <math>1 \leq i \leq k</math>. Find, in terms of <math>n</math> and <math>\ell</math>, the largest real number <math>c</math> such that the inequality
 
Let <math>n>2</math> be an integer and let <math>\ell \in\{1,2, \ldots, n\}</math>. A collection <math>A_1, \ldots, A_k</math> of (not necessarily distinct) subsets of <math>\{1,2, \ldots, n\}</math> is called <math>\ell</math>-large if <math>\left|A_i\right| \geq \ell</math> for all <math>1 \leq i \leq k</math>. Find, in terms of <math>n</math> and <math>\ell</math>, the largest real number <math>c</math> such that the inequality
 
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Latest revision as of 15:38, 15 February 2025

Continued

Let $n>2$ be an integer and let $\ell \in\{1,2, \ldots, n\}$. A collection $A_1, \ldots, A_k$ of (not necessarily distinct) subsets of $\{1,2, \ldots, n\}$ is called $\ell$-large if $\left|A_i\right| \geq \ell$ for all $1 \leq i \leq k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality \[\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2\] holds for all positive integers $k$, all nonnegative real numbers $x_1, \ldots, x_k$, and all $\ell$-large collections $A_1, \ldots, A_k$ of subsets of $\{1,2, \ldots, n\}$. Note: For a finite set $S,|S|$ denotes the number of elements in $S$.

See Also

2024 USAMO (Problems • Resources)
Preceded by
Problem 5 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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