be an acute triangle 
)
and orthocenter 
be the midpoint of 
.
intersects 
,
intersects 
.
and 
lie on 
and 
is a parallelogram.
is tangent to the circumcircle of triangle 
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
without passing through 
be real numbers satisfying![\[
x_1 + x_2 + \cdots + x_6 = 6,
\]](http://latex.artofproblemsolving.com/6/2/2/622a329112ce9613a5c389b4c9078cd43c1c3f05.png)
![\[
x_1^2 + x_2^2 + \cdots + x_6^2 = 18.
\]](http://latex.artofproblemsolving.com/8/f/a/8fa3f74ba7c3587576116a7bdbb4e9f88d389133.png)
Find the maximum possible value of the product![\[
x_1 x_2 x_3 x_4 x_5 x_6.
\]](http://latex.artofproblemsolving.com/3/b/4/3b48315e2775b4ac3913d7c9b02dbc81daed8d15.png)
.
be the midpoint of the arc 
not containing 
similarly. Show that the orthocenter of 
is the incenter 
of 
. Prove that
can be written in the form![\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]](http://latex.artofproblemsolving.com/9/e/9/9e96c07ea3b0750af8bed344df6e8a86fcf0d772.png)
where 
and 
are positive integers. Find the value of 
.
be an acute triangle with 
,
.
be the altitude from vertex 
.
and 
be the feet of the perpendiculars from 
to the lines 
and 
,
intersect the circle 
and 
such that 
.
be the midpoint of segment 
intersect line 
at point 
.
.
is parallel to line 
.
be the orthocenter of triangle 
,
be the midpoint of segment 
is the center (HBC). Let point 
lie on line 
such that ray 
is the angle bisector of 
.
intersect at a point lying on circle 
(i.e., the square on 
has 
,
,
to the centers of the opposite squares are concurrent.
be positive integers. Maximize 
if 
.
,
,
.
Y by doxuanlong15052000, Adventure10
Given triangle 
and let 
be the foot of angle bisectors of 
,respectively.
lie on 
such that 
. Let 
be the foot of 
-altitude on 
.
Points
lie on such that triangles 
are correspondingly similiar (with the given order of vertices) such that 
and 
.
Show that:
and let 
be the foot of angle bisectors of 
,respectively.
lie on 
such that 
. Let 
be the foot of 
-altitude on 
.Points
lie on such that triangles 
are correspondingly similiar (with the given order of vertices) such that 
and 
.Show that:
let 
the proyections of 
at 
a point at 
such that 
is tangent to 
,
is cyclical 
and 
and 
points at 
and 
and 
is evident that 
foot of 
with 
and 
intersection of 
with 
the point on 
and 
reflection of 
about 
.
intersection of 
we can see that 
so 
are cyclic .
and we're done 
