AoPS Academy
, where 
, then the number of integers in the range of 
is __
is inscribed in circle 
. Let 
denote its incenter and 
its 
-excenter. Let 
denote the midpoint of arc 
. Line 
meets a second time at 
. The perpendicular to 
at meets sides 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
, and the circumcircle of 
meets 
a second time at 
. Prove that 
passes through the antipodal to on .
be the center of the circle circumscribed to the triangle 
in which the side 
is parallel to the diameter 
. If 
is the orthocenter of the triangle , prove that the line passes through the midpoint of the segment ![$[OH]$](http://latex.artofproblemsolving.com/7/b/8/7b812c672802ddc509741f60b6fd241be97668af.png)
.
be the center of the circle circumscribed to the triangle in which the side is parallel to the diameter . If is the orthocenter of the triangle , prove that the line passes through the midpoint of the segment .
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
satisfying 
.
is the midpoint of 
.
,