AoPS Academy
+1 w
chessboard into 
non-overlapping rectangles subject to the following conditions:
is the number of white squares in the 
-th rectangle, then 
. for which such a decomposition is possible. For this value of , determine all possible sequences 
.
+1 w
denote the determinant of a square matrix 
. Let ![$g:\left[0, \frac{\pi}{2}\right] \rightarrow R$](http://latex.artofproblemsolving.com/5/3/4/534fcbbc20e62de2dfa2579cd0642663c9fc3380.png)
be the function defined by
where 
.
be a quadratic polynomial whose roots are the maximum and minimum values of the function 
and 
. Then, which of the following is/are TRUE?
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
angles 
and 
are equal 
. On the sides 
and points 
and 
are selected so that 
. It turned out that the line 
is parallel to the angle bisector of the angle . Find the angle 
.
angles and are equal . On the sides and points and are selected so that . It turned out that the line is parallel to the angle bisector of the angle . Find the angle .
through parallel to 
. Let intersect 
at . Notice, that has to be bisector of angle and, as a consequence, of angle in 
. Thus, triangles 
and 
are equal by angles and common side. So 
is parellelogram 
