. Prove that: 
be a triangle, and let 
be a point on side 
.
and ray 
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 
,
so that 
.
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
Given two circles 
and 
intersect at 
and 
. Let 
and 
be two lines through and be symmetric with respect to 
. The line cuts and at 
(
), respectively, the line cuts and at 
(), respectively, such that 
is between 
and 
is between 
. Let 
be the intersection of 
and 
. The line 
cuts 
at 
(
), respectively. Let 
be the intersection of 
and 
. Prove that the circle 
is tangent to .
and 
intersect at 
and 
. Let 
and 
be two lines through and be symmetric with respect to 
. The line cuts and at 
(
), respectively, the line cuts and at 
(), respectively, such that 
is between 
and 
is between 
. Let 
be the intersection of 
and 
. The line 
cuts 
at 
(
), respectively. Let 
be the intersection of 
and 
. Prove that the circle 
is tangent to .This post has been edited 1 time. Last edited by parmenides51, Jul 27, 2020, 9:44 PM
and 
is cyclic. The rest is smooth.
