,
is the midpoint of 
,
the point of intersection of the tangents at points 
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
A point 
and two congruent squares 
and 
are given on the plane of area 
. Figure 
is formed by all such points 
that the vector 
can be written as 
. where point 
belongs to the square , and point 
belongs to the square 
. Find the smallest and the largest possible values of the area of the figure depending on the relative position of the squares and .
and two congruent squares 
and 
are given on the plane of area 
. Figure 
is formed by all such points 
that the vector 
can be written as 
. where point 
belongs to the square , and point 
belongs to the square 
. Find the smallest and the largest possible values of the area of the figure depending on the relative position of the squares and .This post has been edited 3 times. Last edited by parmenides51, Jul 26, 2021, 3:10 PM
