be an acute triangle with 
,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
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 
.
Y by Adventure10, Mango247
The sides ![$ [AB]$](//latex.artofproblemsolving.com/d/7/a/d7a8027c238eec9cf67de0f7ec6cb1df4df49a61.png)
and ![$ [AC]$](//latex.artofproblemsolving.com/5/b/0/5b08b2b92472414250205866b93405af4772e86c.png)
of the triangle 
are tangent to the incircle with center 
of the 
at the points 
and 
, respectively. The internal bisectors of the drawn form 
and 
intersect the line 
at the points 
and 
, respectively. Suppose that 
is the intersection point of the lines 
and 
. Prove that 
.
![$ [AB]$](http://latex.artofproblemsolving.com/d/7/a/d7a8027c238eec9cf67de0f7ec6cb1df4df49a61.png)
and ![$ [AC]$](http://latex.artofproblemsolving.com/5/b/0/5b08b2b92472414250205866b93405af4772e86c.png)
of the triangle 
are tangent to the incircle with center 
of the 
at the points 
and 
, respectively. The internal bisectors of the drawn form 
and 
intersect the line 
at the points 
and 
, respectively. Suppose that 
is the intersection point of the lines 
and 
. Prove that 
.
.
is the intersection of 
and 
,
,
;
follows by symmetry.
be on 
such that 
.
intersect 
;
is the angle bisector of 
,
and 
,
and line 
.![\[ \frac{AM}{MB} \cdot \frac{BQ}{QB'} \cdot \frac{NB'}{NA} = 1\]](http://latex.artofproblemsolving.com/b/e/4/be4f37b39f5cd524db8a650b77b73212e84b2912.png)
![\[ \Longleftrightarrow \frac{ \frac{b+c-a}{2} }{ \frac{c+a-b}{2} } \cdot \frac{BQ}{QB'} \cdot \frac{ a-\frac{a+b-c}{2} }{ \frac{b+c-a}{2} } = 1\]](http://latex.artofproblemsolving.com/6/7/c/67cc4e14d8a7ace32692a85603214c3d20c058ed.png)
![\[ \Longleftrightarrow \frac{ b+c-a }{ c+a-b } \cdot \frac{BQ}{QB'} \cdot \frac{ c+a-b }{ b+c-a } = 1\]](http://latex.artofproblemsolving.com/8/c/2/8c293903fa061489e88a7914a2e8755ad96e06d7.png)
,
. Notice that 
, then 
,
,
is cyclic, so 
.
.
is the orthocenter of the 
,where 
.
of the 
and 
. Therefore,
is cyclically 
.
is cyclically 
.
and 
obtain that the incenter 
.
, then 
and 
.
