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 
,
its incenter, and the midpoint of the side . If 
, determine the smallest possible value of the angle 
.
since they have SSA and is on the correct side of the altitude as 
is. Thus 
so 
We can see that the converse of this is true by SAS too, so 
Therefore we can treat the length condition as equivalent to 
since any such triangle will have both. The problem is reduced now to minimizing 
with 
so we wish to maximize 
given Then 
Since 
we know 
is acute, so 
We can indeed achieve this maximum of 
with 
and 
A 30-60-90 triangle, in fact. Therefore, the minimum of is 
degrees or 
which we can see works on any 30-60-90 triangle.
is congruent to 
, we can deduce that 
. So we need to maximise 
. lies on the circle centre 
radius . Therefore is maximised when is tangent to this circle, and this is when 
. Since 
, is half of an equilateral triangle so 
. has a minimum of 
degrees.
