,
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
Trapezium 
and parallelogram 
are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part.
(Yu. Blinkov)
and parallelogram 
are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part.(Yu. Blinkov)
This post has been edited 1 time. Last edited by parmenides51, Sep 14, 2020, 12:22 PM
