is the midpoint of 
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 
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 Adventure10, Mango247
Segment 
is the diameter of the circumscribed circle of an acute-angled triangle 
. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side 
, which intersects sides 
and 
at points 
and 
, respectively. Prove that the perimeter of the triangle 
is two times larger than the side .
is the diameter of the circumscribed circle of an acute-angled triangle 
. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side 
, which intersects sides 
and 
at points 
and 
, respectively. Prove that the perimeter of the triangle 
is two times larger than the side .This post has been edited 1 time. Last edited by parmenides51, Feb 25, 2024, 7:39 AM
