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be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle with 
, its orthocenter and 
it's circumcenter. Let be a point on the parallel through to such that 
and and are on different sides of . Denote by 
the intersection of the circumcircle of and 
, where 
is the reflection of over , 
the midpoint of 
, the intersection of 
and the parallel through to 
, the intersection of 
and the perpendicular through to 
and 
a point on such that 
, where 
is the midpoint of 
. Prove that 
lie on a line.
of integers is rootiful if, for any positive integer 
and any 
, all integer roots of the polynomial 
are also in . Find all rootiful sets of integers that contain all numbers of the form 
for positive integers 
and 
.
be an isosceles triangle with 
. Let 
be a point on 
. Let 
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
+1 w
of a triangle 
to its sides 
are denoted 
respectively. Let 
be the inradius of the triangle. Prove that:
we know that ![$[GBC]=[GAB]=[GCB]\implies g_a .a=\frac13 r(a+b+c) \iff g_a=\frac13 r. \frac{a+b+c}{a}$](http://latex.artofproblemsolving.com/5/7/5/575250aa3e20465e3866496d557087b3ae90d26f.png)
but 
then 
idem for the others
but since 
hence 