[/quote]
,
be an odd prime and 
be an integer such that 
but 
.![\[ p \mid (p - 1)!\left(x - \frac {x^2}{2} + \frac {x^3}{3} - \cdots - \frac {x^{p - 1}}{p - 1}\right).\]](http://latex.artofproblemsolving.com/1/a/3/1a36f070b56129d302daa3eb5d6dcb1851a90637.png)
John Berman
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
and real numbers 
,
,
.
be a triangle with circumcircle 
.
and 
intersect at 
.
intersects the line 
at 
and 
is the midpoint of 
.
and 
intersect on 
![\[
\text{Find all prime numbers } p, q \text{ such that } p^2 \mid q^3 + 1 \text{ and } q^2 \mid p^3 + 1.
\]](http://latex.artofproblemsolving.com/3/d/3/3d3bad32fb51a94029f26610f800ed9a74bc583d.png)
,
,![\[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]](http://latex.artofproblemsolving.com/9/3/9/9399f4c397bc3c83aaa518a0fc5b4ce93f84cdce.png)
and 
Prove that
Let 
Prove that
and 
Prove that
Let 
Prove that
and 
Prove that
Let 
Prove that
Prove that
and 
.
Let 
.
Let 
.
Y by
Somehow I broke the unique contest collection naming scheme; for what it's worth, I also ran into an AJAX timeout error while trying to make this particular post collection for the first time, and pressed "create" twice.
Not sure if there's anything to be done here, but are there potentially unforeseen consequences here?
https://artofproblemsolving.com/community/c4258765_2024_puerto_rico_team_selection_test (this is the one I'm working on; it's also not finished yet)
https://artofproblemsolving.com/community/c4258764_2024_puerto_rico_team_selection_test
Not sure if there's anything to be done here, but are there potentially unforeseen consequences here?
https://artofproblemsolving.com/community/c4258765_2024_puerto_rico_team_selection_test (this is the one I'm working on; it's also not finished yet)
https://artofproblemsolving.com/community/c4258764_2024_puerto_rico_team_selection_test
This post has been edited 2 times. Last edited by Equinox8, Yesterday at 5:25 PM
Reason: clarity
Reason: clarity
