[/quote]
,
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
that satisfy the equation
.
at 
.
and 
,
and 
be the incenter of triangles 
and 
,
.
,
,
,
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.
. Prove that: 
and ray 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
is an equilateral . Let 
at 
at 
Y by lahmacun, Adventure10
Let 
be an isosceles triangle with 
. Let 
be the diameter of the circumcircle of and let 
be a point on the smaller arc 
. The line 
intersects the rays 
and 
at points 
and 
, respectively. The line intersects the lines 
and 
at points 
and 
, respectively. Prove that the midpoint of 
lies on the circumcircle of 
be an isosceles triangle with 
. Let 
be the diameter of the circumcircle of and let 
be a point on the smaller arc 
. The line 
intersects the rays 
and 
at points 
and 
, respectively. The line intersects the lines 
and 
at points 
and 
, respectively. Prove that the midpoint of 
lies on the circumcircle of 
is just a reflection of 
.
.
and 
are cyclic.
.
.
and 
intersect 
is the angle bisector of 
.
.
.
and 
.
is a rectangle.
at 
and 
are both perpendicular to 
.
.
is perpendicular bisector of 
.
is cyclic
is cyclic
.
,
.
.
is perpendicular bisector of 
.
is midline in 
,
.
which implies that 
is cyclic and we are done. 
