. Prove that: 
be a triangle, and let 
be a point on side 
.
at 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
such that 
for all 
,
.
and 
on the plane such that 
is an equilateral . Let 
and 
at 
and 
at 
,
and 
.
and 
is equal to 
.
over 
.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
Prove that
of the pizza, where 
is a positive integer. Find 
Y by Adventure10, Mango247
Let 
be an acute-angled triangle of circumcenter 
and orthocenter 
. Let 
be the midpoint of 
be the symmetric of with respect to 
be the midpoint of 
and 
be a point on the line A H such that 
is parallel to 
. Prove that 
and 
are perpendicular.
be an acute-angled triangle of circumcenter 
and orthocenter 
. Let 
be the midpoint of 
be the symmetric of with respect to 
be the midpoint of 
and 
be a point on the line A H such that 
is parallel to 
. Prove that 
and 
are perpendicular.
,
is perpendicular to 
with respect to 
,
is a parallelogram. Then note that 
is the midline of 
,
is also a parallelogram. And since 
(a well-known result) 
,
as desired 
