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)
of the pizza, where 
is a positive integer. Find .
, the angle 
is 
. Point is the midpoint of side . Point 
is symmetric to point wrt side . The segments AP and BC meet at point 
. Lines 
and intersect at point 
. What is the ratio of the segments 
?
, the angle is . Point is the midpoint of side . Point is symmetric to point wrt side . The segments AP and BC meet at point . Lines and intersect at point . What is the ratio of the segments ?
.
and 
.
cuts the x-axis in 
. and cut in 
.
.
and 
Prove that
such that 
for all 
Prove that
