Difference between revisions of "2024 USAMO Problems/Problem 6"
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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== | ||
| + | ==See Also== | ||
| + | {{USAMO newbox|year=2024|num-b=4|num-a=6}} | ||
| + | {{MAA Notice}} | ||
Revision as of 17:21, 2 June 2024
Let 
be an integer and let 
. A collection 
of (not necessarily distinct) subsets of 
is called 
-large if 
for all 
. Find, in terms of 
and , the largest real number 
such that the inequality
![\[\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2\]](http://latex.artofproblemsolving.com/3/e/d/3ed5047ee81526de349e5b0029f19c0187c25429.png)
holds for all positive integers 
, all nonnegative real numbers 
, and all -large collections of subsets of .
Note: For a finite set 
denotes the number of elements in 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
