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 
,
, points 
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
The triangle 
has three acute angles. Its altitudes 
and 
are extended to meet the circle circumscribed to the triangle , respectively at 
and 
. What is the value of 
:
a) If the triangle is equilateral?
b) If the triangle has any three acute angles?
has three acute angles. Its altitudes 
and 
are extended to meet the circle circumscribed to the triangle , respectively at 
and 
. What is the value of 
:a) If the triangle
is equilateral?b) If the triangle
has any three acute angles?This post has been edited 1 time. Last edited by parmenides51, Nov 20, 2020, 10:11 PM
