and 
.
Let 
.
Prove that
and 
Prove that
'is a perfect square.
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
of an equilateral triangle 
are taken points 
and 
,
, points 
and 
so that 
.
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
Let 
be a convex quadrilateral with 
. Let the incircles of triangles 
and 
touch 
at 
and 
, respectively, with lying in between 
and . If 
and 
then what is the sum of the radii of the incircles of triangles and 
?
be a convex quadrilateral with 
. Let the incircles of triangles 
and 
touch 
at 
and 
, respectively, with lying in between 
and . If 
and 
then what is the sum of the radii of the incircles of triangles and 
?
as 
.
.
by equal tangents. Since 
,
.
due to the triangle being right. Thus our answer is 
