AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
that satisfy the condition 
for the natural number 
Here 
and 
are natural numbers. Is there a natural number 
such that each of the numbers 
is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
we have 
.
. has orthocenter . Let 
be a point distinct from the vertices on the circumcircle of . Suppose that circle 
meets 
at 
, and circle 
meets 
at 
. Prove that as moves on the circumcircle, the reflection of across line 
also moves on a fixed circle. there exists an -digit number divisible by 
all of whose digits are odd.
and 
.
?
chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes 
rows and columns and looks at the 
cells in the intersection. Babek wants to have all these cells in a same color, but Azer doesn't. With at least how many colors, Azer can reach his goal?
,
and 
. Interior angle bisector of 
intersects 
at 
, and 
at 
.Prove that circumcircle of 
passes from circumcenter of 
.
with 
and let 
be the midpoint of 
side. 
is chosen in such that 
. Suppose 
. Prove that 
.
such that
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
.
.
.
which means 
which means 
.
,