[/quote]
,
,
the point of intersection of the tangents at points 
of the circumscribed circle of 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
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)
such that 
that satisfy the equation
.
.
and 
,
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 
.
.
Y by Adventure10
Two circles 
and 
are internally tangent at the point 
. The chord 
of is tangent to at the point 
, and the segments 
and 
intersect at the points 
and 
. Let 
and 
be the reflections of and about the line 
; and let 
and 
be the reflections of and about the line . The lines 
and 
intersect at the point 
.
Prove that the lines
and are perpendicular.
and 
are internally tangent at the point 
. The chord 
of is tangent to at the point 
, and the segments 
and 
intersect at the points 
and 
. Let 
and 
be the reflections of and about the line 
; and let 
and 
be the reflections of and about the line . The lines 
and 
intersect at the point 
.Prove that the lines
and are perpendicular.
and 
are collinear as 
.
,
are collinear.
and similar for the other one 
i
is cyclic
and 
from Claim 4 and hence 
