[/quote]
,
,
the point of intersection of the tangents at points 
and 
.
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 
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 
.
.
Y by Adventure10, Mango247
Let 
be an acute triangle such that 
. The circle with diameter 
intersects the internal angle bisectors of 
and 
at the points 
and 
, respectively 
. The internal bisector of 
intersects 
and at the points 
and 
, respectively. Prove that 
.
be an acute triangle such that 
. The circle with diameter 
intersects the internal angle bisectors of 
and 
at the points 
and 
, respectively 
. The internal bisector of 
intersects 
and at the points 
and 
, respectively. Prove that 
.
be the altitudes of 
.
are similar with ratio 
then
