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 orthocenter 
,
and that line 
is not parallel to line 
.
is perpendicular to 
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
Two circles 
and 
touch externally at 
. their external common tangent touches at 
and at 
. A line, parallel to 
and tangent to at 
, intersects at 
and 
. Prove that points 
are collinear.
(A. Kalinin)
and 
touch externally at 
. their external common tangent touches at 
and at 
. A line, parallel to 
and tangent to at 
, intersects at 
and 
. Prove that points 
are collinear.(A. Kalinin)
This post has been edited 1 time. Last edited by parmenides51, Jul 29, 2018, 7:52 PM
Reason: source typo
Reason: source typo
and point 
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and 
,![\[\angle HFC=\angle HCF=\angle HCA=\angle CAG=\angle AEF=\angle AFG\]](http://latex.artofproblemsolving.com/3/f/3/3f37977eb9ff1bf1b90c26b14a3ab9612bda4ded.png)
which means 
.
.
.
and 
since they are tangents. Then we see that points 
