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.
with side length of 
. The point 
is selected on the diagonal 
such that 
. Denote the circumcenters of 
as 
. Find the length of 
on a circle and points 
and 
on segment , such that 
Find the ratio 
, where 
is a point on arc , such that 
is maximized.
such that 
. Denote the midpoints of 
as 
respectively, compute the area of 
given the area of is 
.
such that for all real numbers 
and 
,
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 
.
be distinct real numbers such that 
and 
Evaluate all the possible values of 
.
be an acute scalene triangle with orthocenter 
. Line 
intersects 
at 
and line 
intersects 
at 
. Let 
be the foot of the perpendicular from to the line through 
parallel to 
. Point 
lies on line 
such that 
is parallel to , and point 
lies on line 
such that 
is parallel to . Prove that points 
, 
, , are concyclic.
that is inscribed in 
. The incircle 
is tangent to 
at 
respectively. A line through 
parallel to intersects at 
, and 
intersects again at 
. Let 
is the midpoint of . 
be the second intersection of 
with . 
intersects again at . Prove that the line passing through perpendicular to 
bisects 
.
![\[\left(-\cos(1^{\circ})\right)\left(-\cos(2^{\circ})\right)\left(-\cos(4^{\circ})\right) \cdots \left(-\cos{(256^{\circ})}\right)\]](http://latex.artofproblemsolving.com/4/4/e/44e5bbdf821aed7bb2d855efcf39af763ebaea48.png)
on a blackboard. Then, he erases each term on the blackboard with probability 
. What is the expected value of the remaining expression?
It can't seem to be simplified further.
i thought that after you could try multiplying by 
but it doesn't work 
.
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
=2$$](http://latex.artofproblemsolving.com/c/8/b/c8b33f21c28cd9d193abb2848116835353aeb8bd.png)
