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.
such that 
.
is selected on arc 
not containing 
on 
. 
and 
are the reflections of point 
and 
with respect to 
, respectively. Circumcircles of 
and 
intersect at 
. Prove that 
is colinear
has 
vertices and 
edges, where 
, show that there exists a path with 
vertices. is good if 
for some positive integer 
. Here 
is the sum of all digits of . Prove that:
be a triangle with circumcenter 
, angle bisector with 
and altitude 
with 
. The lines 
and meet at 
. The circumcircle of 
meets 
at 
and 
meets at 
. The circumcircles of triangles 
and meet at 
. Show that 
is a symmedian in 
+1 w
so that no two exist that their difference equals to 
.
, is a point on such that is the internal bisector of 
. Now Suppose 
=
and 
. Prove that 
.
be n positive real numbers such:
.then prove that :
. actually this problem was a problem from morracan tst.
.
for some fixed 
is homographic and has standard solution. More precisely, in this case 
satisfies a second order recurrence relation of the form 
, which can be solved and after that it's just boring computation.
for some fixed is homographic and has standard solution.
be an increasing sequence of positive integer such that 
.
there exists 
Prove that
Where 
and 
Prove that
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
,
,
be 
.
.
denote 
.