be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
with diameter 
is given. The point 
intersects 
and 
lies on 
is tangent to 
.
,
colours (so that adjacent vertices have different colours).
edges can be removed from the graph so that it remains connected.
be the incenter and the circumcenter of triangle 
be the circumcenters of triangle 
.
be the midpoints of segments 
.
,
.
Prove that
E
matrix equals either 
or 
Let 
denote the sum of eight products of entries in each row. Also, let 
denote the sum of eight products of entries in each column. Find the maximum possible value of 
In other words, find![$$ {\rm max}\ \left[ \sum_{i=1}^8\ \prod_{j=1}^8\ a_{ij} -
\sum_{j=1}^8\ \prod_{i=1}^8\ a_{ij} \right]
$$](http://latex.artofproblemsolving.com/2/d/e/2de7fd40366073445d289356d0f4e66b542cb60c.png)
matrix.
Show that there exists a constant 
such that for all positive integer 
![\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]](http://latex.artofproblemsolving.com/f/e/9/fe933e886e28c2ecf1fa880d0e650ac7a2998aaa.png)
Proposed by Cheng Jiang
is written on the board. A move consists of adding either 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
,
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
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!
