[/quote]
,
Prove that
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
and 
.
Let 
.
such that 
that satisfy the equation
with incenter 
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
is the reflection of 
.
be the reflection of 
.
concyclic.
and 
Prove that
. Prove that: 
Y by Adventure10
Two circles, 
and 
, intersect at 
and 
. Let 
a point in the arc 
in . Lines 
and 
cut back to in 
and 
, respectively. Let 
be another point in the same arc as , so that lines 
and 
re-intersect in 
and 
, respectively. Prove that the arcs 
and 
have equal measure.
and 
, intersect at 
and 
. Let 
a point in the arc 
in . Lines 
and 
cut back to in 
and 
, respectively. Let 
be another point in the same arc as , so that lines 
and 
re-intersect in 
and 
, respectively. Prove that the arcs 
and 
have equal measure.
