AoPS Academy
, with 
as its incenter and 
as its circumcircle, 
intersects again at 
. Let 
be a point on the arc 
, and 
a point on the segment 
, such that 
. If 
is the midpoint of 
, prove that the meeting point of the lines 
and 
lies on .
be reals such that 
and 
Find tha minimum value of 
be a triangle with the incenter , and let the incircle 
of touch 
at points 
respectively. Let 
intersect again at a point 
. Let the lines through and parallel to 
intersect again at points 
respectively. Prove that lies on the common chord of the circumcircle of 
and the circumcircle of 
.
. Prove that:
be a point in the plane, and 
a line not passing through . Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. over ? to ? be a triangle with 
and with its incircle touching the sides 
and at 
and 
respectively. A point lies on the extension of beyond 
such that 
. Let 
be the midpoint of 
. Prove that the points , , are collinear.
be a triangle with 
and 
.
inside triangle satisfies the following conditions:![\[
XA^2 + BC^2 \leq XC^2 + AB^2 \leq XB^2 + AC^2.
\]](http://latex.artofproblemsolving.com/0/3/1/031aa58d7d3dabe636dfb4245ef2456a2a3d5df6.png)
Find the maximum value of 
such that![\[
\frac{XA}{AC} \geq m.
\]](http://latex.artofproblemsolving.com/c/5/e/c5ed2892b224aff3b2f3e20b4c9424707ff88690.png)
be reals such that 
Prove that
Show that 
.![$$ \frac{ a}{b}+\frac{b}{c}+\frac{c}{a}\geq \sqrt[3]{49}$$](http://latex.artofproblemsolving.com/4/1/7/41747f09ab60b8e1a07de01f23d11985133c0d17.png)
.
