the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
and 
Prove that
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
and 
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)
Y by
In triangle 
, the bisector 
, 
, intersects the circumcircle, 
, of triangle at point 
. Perpendicular from 
on 
intersects the circle at point 
. The line 
intersects the side 
at the point 
. Show that 
is perpendicular to 
.
, the bisector 
, 
, intersects the circumcircle, 
, of triangle at point 
. Perpendicular from 
on 
intersects the circle at point 
. The line 
intersects the side 
at the point 
. Show that 
is perpendicular to 
.This post has been edited 1 time. Last edited by parmenides51, Aug 17, 2020, 7:14 PM
