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

(Created page with "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>\...")
 
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holds for all positive integers <math>k</math>, all nonnegative real numbers <math>x_1, \ldots, x_k</math>, and all <math>\ell</math>-large collections <math>A_1, \ldots, A_k</math> of subsets of <math>\{1,2, \ldots, n\}</math>.
 
holds for all positive integers <math>k</math>, all nonnegative real numbers <math>x_1, \ldots, x_k</math>, and all <math>\ell</math>-large collections <math>A_1, \ldots, A_k</math> of subsets of <math>\{1,2, \ldots, n\}</math>.
 
Note: For a finite set <math>S,|S|</math> denotes the number of elements in <math>S</math>.
 
Note: For a finite set <math>S,|S|</math> denotes the number of elements in <math>S</math>.
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==See Also==
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==See Also==
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{{USAMO newbox|year=2024|num-b=4|num-a=6}}
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{{MAA Notice}}

Revision as of 16:21, 2 June 2024

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

See Also

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

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