AoPS Academy
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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 
.
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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.
be a triangle with 
. Let be a point on the line 
beyond 
such that 
. Let be the mid-point of and let 
be the point on the side such that 
. Prove that 
be a point on the line , above such that 
=
is the midpoint of side 
,
is parallel to 
(By midpoint theorem's converse)
as 
and angle chasing
=
=
is isosceles
which gives 
> allows us to construct the desired line-segment and finish the problem
is isosceles.
to 
.
.
.
.
.
(exterior angle).
,
.
.