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be an acute triangle with 
, and let 
be the projection from 
onto 
. Let 
be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across 
and 
be 
and 
, respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at 
and 
. Let 
intersect at 
. Prove that 
.
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 
.
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?
(
) has the property that there are points and on sides and respectively such that 
and 
. Show that the points and are equidistant from the intersection point of the diagonals of the trapezoid.
() has the property that there are points and on sides and respectively such that and . Show that the points and are equidistant from the intersection point of the diagonals of the trapezoid.
and 
are cyclic
that satisfy the angle condition. Let 
. Using Reim's, we see that is also cyclic. Then, 
meaning that 
. 
be the intersection of the diagonals and assume that line parallel to and 
passing through intersects at 
and at .
.
Notice that using Reim's 
is also cyclic. is the center of 
.
bisects the angle 
. Let 
. Then, using sines theorem, 
Since 
, we see that 
Using Claim 3 it is clear that 
and thus, we are done. 
is the reflection of 
and