AoPS Academy
+1 w
[/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.
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
with incenter 
, the incircle 
touches sides 
and 
at points 
and 
, respectively. A circle passing through 
and 
touches at point 
. The circumcircle of 
meets 
at 
. Prove that 
is parallel to 
.
with incircle . Let 
be the point on such that the circumcircle of 
is tangent to and let the 
-excircle be tangent to at 
. Prove that the tangent from to and the tangent from to (distinct from ) meet on the line parallel to and passing through .
and 
Prove that
Let and 
Prove that
Let and 
Prove that
with side length of 
. The point 
is selected on the diagonal such that 
. Denote the circumcenters of 
as 
. Find the length of 
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
such that 
. Denote the midpoints of 
as 
respectively, compute the area of 
given the area of is 
.
with 
. Point lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
are on the same line, compute the length of .
where 
and 
.
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
where 
is the number of lines.