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 triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
fair coins are flipped. If 
of them land heads, find the expected value of 
.
satisfying
Prove
and 
be positive integers. Suppose that the sum of integers between and , including and , are equal to 
.
, find the pair such that achieves the minimum.
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
be a triangle with circumcircle . Let 
and 
be two lines through the points 
and 
, respectively, such that 
. The second intersections of and with are and 
, respectively. Assume that and are on the same side of as . Let 
intersect at 
and let 
intersect at 
. If 
, 
and 
are circumcenters of the triangles , 
and 
, respectively, and is the circumcenter of the triangle 
, prove that 
.
is . Two moving points 
moves on sides 
, satisfy that 
. Prove that 
is a definite value.
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
.
Where 
because the main diagonals 
diagonals
