,
,
is __
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 inscribed in circle 
.
denote its incenter and 
its 
-
.
meets 
.
at 
and 
at 
respectively. The circumcircle of 
meets 
a second time at 
meets 
a second time at 
.
passes through the antipodal to 
Y by
Let 
be a cyclic trapezium with 
and 
. The circle is called 
, and has centre 
. Let 
be a variable point on the part of the ray 
that is beyond 
. It is given that 
is not tangent to * . The circle with diameter 
meets again at 
. Let 
be the intersection of the lines and 
, and 
be the second point of intersection of the line and . Prove that the lines 
pass through a fixed point as varies.
* The official wording had the comment: (GCS: I don’t see how it could be, but that is what the question says!)
be a cyclic trapezium with 
and 
. The circle is called 
, and has centre 
. Let 
be a variable point on the part of the ray 
that is beyond 
. It is given that 
is not tangent to * . The circle with diameter 
meets again at 
. Let 
be the intersection of the lines and 
, and 
be the second point of intersection of the line and . Prove that the lines 
pass through a fixed point as varies.* The official wording had the comment: (GCS: I don’t see how it could be, but that is what the question says!)
be the reflection of 
be the foot from 
to 
are collinear. Call 
the circle with diameter 
the circle with diameter 
.
.
