chessboard into 
non-overlapping rectangles subject to the following conditions:
is the number of white squares in the 
-
.
.
denote the determinant of a square matrix 
.![$g:\left[0, \frac{\pi}{2}\right] \rightarrow R$](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 
.
with circumcenter 
,
and that line 
is not parallel to line 
.
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
Y by Adventure10, Mango247
A point 
is marked on a chord 
of a circle 
Let 
be the midpoint of 
and 
be the center of the circle The circumcircle of the triangle 
intersects the circle 
again at point 
and the straight line 
again at point 
Prove that the circumcircle of the triangle 
touches 
is marked on a chord 
of a circle 
Let 
be the midpoint of 
and 
be the center of the circle The circumcircle of the triangle 
intersects the circle 
again at point 
and the straight line 
again at point 
Prove that the circumcircle of the triangle 
touches 
over 
.
,
which implies 
.
is the midpoint of 
,
.
,
