AoPS Academy
'is a perfect square.
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 
intersect at points 
and 
. A common tangent is drawn that touches the circles and respectively at the points 
and 
. Prove that the triangles 
and 
are equal.
and 
.
Let 
.
Prove that
and 
Prove that
such that 
for all 
is humpty point in triangle 
and 
are equidistant from the line ![$[AB {{C} _{1}}]=[AB {{C} _{2}}]$](http://latex.artofproblemsolving.com/8/a/a/8aaa4ee911e1bc6b901b5a3392fc49ec51347c36.png)
.
is congruent to 
?