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 an acute angled triangle with incenter . Line perpendicular to 
at meets 
and 
at points and respectively. Let 
be the incenters of 
and 
respectively. Suppose 
lie on a circle. Prove that 
.
be the reflection of 
upon . Clearly, 
, 
, therefore 
is an isosceles trapezium which means 
all lie on a circle, which is absurd since obviously 
is a concave quadrilateral. Thus 
. And so, 
. And we are done. 
be the reflection of upon . Clearly, , , therefore is an isosceles trapezium which means all lie on a circle, which is absurd since obviously is a concave quadrilateral. Thus . And so, . And we are done. ![[asy]
pair D=dir(-110),Dp=dir(-70),P=dir(-160),Q=dir(-20),E=dir(30),Ep=dir(150),B=extension(Ep,D,Dp,E);
draw(unitcircle,magenta);
dot("$D$",D,dir(D));
dot("$E$",E,dir(E));
dot("$E'$",Ep,dir(Ep));
dot("$D'$",Dp,dir(Dp));
dot("$P$",P,dir(P));
dot("$Q$",Q,dir(Q));
dot("$B$",B,dir(B));
draw(Ep--B--E,royalblue);
[/asy]](http://latex.artofproblemsolving.com/c/7/0/c7051d09808d608c08e563ec9a1984c618c58aa6.png)
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 
