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 inscribed in a rhombus ABCD. points E,F,G,H are chosen on sides AB, BC, CD, DA respectively so that EF and GH are tangent to inscribed circle. show that EH and FG are parallel.
![[asy]
size(10cm);
pair X = dir(135);
pair Y = dir(45);
pair B = extension(X,X+dir(90)*X,Y,Y+dir(90)*Y);
pair A = extension(B,X,dir(0),dir(180));
pair C = extension(B,Y,dir(0),dir(180));
pair Z = dir(100);
pair E = extension(A,B,Z,Z+dir(90)*Z);
pair F = extension(B,C,Z,Z+dir(90)*Z);
pair O = (0,0);
draw(A--B--C--A--cycle);
draw(arc((0,0),1,0,180));
draw(E--F);
draw(X--O--E);
draw(Z--O--F);
draw(O--Y);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$E$",E,dir(E));
dot("$F$",F,dir(F));
dot("$X$",X,dir(X));
dot("$Y$",Y,dir(Y));
dot("$Z$",Z,dir(Z));
dot("$O$",O,dir(270));
[/asy]](http://latex.artofproblemsolving.com/1/a/b/1ab995a13a0e1fc2e5264c623c74fa8487d24f04.png)
, or that 
. Thus, it suffices to show that 
.
be the tangency points of the circle to 
and 
respectively. Note that 
, 
and 
. Therefore,![\[\angle AEO=90^\circ - \angle XOE=\angle FOC.\]](http://latex.artofproblemsolving.com/9/7/2/972004e78864ca2560ccaa07555112c68d821238.png)
Hence, the triangles 
and 
are similar. It follows that 
. Similarly, 
. The conclusion follows.
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 
