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)
is irrational?
such that 
for all 
![[asy]
pair A,B,C,D,E,F,G,H;
A=(0,0);
B=(7,0);
C=(0,7);
D=(4,3);
E=(0,3);
F=(2,1);
G=(1,0);
H=(0,1);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
draw(F--G--H--cycle);
[/asy]](http://latex.artofproblemsolving.com/9/e/2/9e2ba059fa26ebd115ed00773cebd9d977fd0a4c.png)