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such that 
, for all 
.
is a set of positive integers.)
touch sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point. is on side of triangle 
. Let 
, 
and 
be nine-point centers of triangles 
, 
and respectively. Prove that perpendicular lines from to 
, from to , from to and from 
to are concurrent.
of real numbers that satisfy the system of equations![\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]](http://latex.artofproblemsolving.com/d/4/d/d4d6c485b4653577e732ff235448bb779914a576.png)
be a convex quadrilateral. Let 
be points on sides , , 
respectively such that:
.
be the intersection of and 
.
.
so 
,
,
,
.