[/quote]
,
and 
Prove that
,
,
,
are integers such that 
.
and 
of 
such that for each integer 
with 
,![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be an odd prime and 
be integers so that the integers 
are divisible by 
Prove that 
be real numbers such that 
Prove that 
Where 
be a triangle with circumcircle 
.
and 
respectively denote the midpoints of the arcs 
and 
that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc 
including 
)
be the incenter of 
be the circle that is tangent to 
be the circle that is tangent to 
,
friends, 
people are seated around a circular table in a fixed order. Ali places 
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
is the circumcenter of 
.
lie on 
,
is tangent to circumcircle of 
.
is tangent to circumcircle of triangle 
.
Y by evt917
When I had less than 25 posts on AoPS, I saw many people create threads about them getting 1000th posts. I thought I would never hit 1000 posts, but here we are, this is my 1000th post.
As a lot of users like to do, I'll write my math story:
In conclusion, AoPS has helped me improve my math. I have also made many new friends on AoPS!
Finally, I would like to say thank you to all the new friends I made and all the instructors on AoPS that taught me!
As a lot of users like to do, I'll write my math story:
In conclusion, AoPS has helped me improve my math. I have also made many new friends on AoPS!
Finally, I would like to say thank you to all the new friends I made and all the instructors on AoPS that taught me!
I hope you are doing well and advancing in the AoPS community!
