AoPS Academy
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 .
, where the common ratio 
. Given that
.
. First recall the following fact.
and ratio 
then the sum is given by 
Thus we have that 
As 
thus 
and the finish is easy.
as 
)
Now we just have to find the range of 
as ranges over 
. Over 
the expression takes values in the interval 
, and at 
it takes the value 
, and nowhere else does it take this value. So the answer is 
. 
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
