AoPS Academy
, point 
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 
. The segment 
meets the circumcircle at the point 
. Prove that 
.
and 
be positive real numbers with 
. Prove that
For which and equality holds?
, find the number of terms with coefficient divisible by 
.
and of an equilateral triangle are taken points 
and 
, and on the side 
, points 
and so that 
. Prove that the angle between lines 
and 
is equal to 
.
over 
. Ignore "pathological" solutions.[/quote]
![\[\lfloor x \rfloor\]](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)
and lie on the hypotenuse of a right triangle . Prove that orthocenters the triangles 
and 
coincide if and only if 
and lie on the hypotenuse of a right triangle . Prove that orthocenters the triangles and coincide if and only if 
:
and 
Prove that
such that 
for all 
