AoPS Academy
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
, 
, 
be real positive numbers. Show that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]](http://latex.artofproblemsolving.com/0/9/5/0959464bbb9b821a11df292f9e6addded7326371.png)
fair coins are flipped. If 
of them land heads, find the expected value of 
.
and 
be given circles. Let 
on and 
on be fixed points. If chord 
of is parallel to chord 
of , find the locus of the midpoint of 
.
be the centers of 
Let 
be the midpoints of 
and 
the midpoints of 
i.e. the projections of on 
respectively. Let 
be the midpoint of 
and the perpendicular bisector 
of cuts 
at 
respectively. Since 
is midparallel of 
then is midpoint of 
and 
which means that 
and 
are equidistant from goes through the midpoint 
of 
locus of 
is the circle with center and radius 
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
and 
.
Let 
.
Prove that
