is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
be the circumcircle of a non-isosceles triangle 
be a tangent line to 
)
be midpoint of 
.
be its circumcenter and 
its incenter.
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of 
is parallel with line 
.
such that, for all positive integers 
and 
,
Y by Adventure10, Mango247
Let 
be an acute-angled triangle with 
. Tangent to its circumcircle 
at 
intersects the line 
at 
. Let 
be the centroid of 
and let 
meet again at 
. Suppose the line 
intersects the lines 
and 
at 
and 
, respectively. Prove that 
.(Slovakia)
be an acute-angled triangle with 
. Tangent to its circumcircle 
at 
intersects the line 
at 
. Let 
be the centroid of 
and let 
meet again at 
. Suppose the line 
intersects the lines 
and 
at 
and 
, respectively. Prove that 
.(Slovakia)
be the line parallel to 
.
.
from 
.
.
.
maps the line 
to 
.
and the midpoint of 
.
.
the line parallel to 
the point in 
and let 
.
the midpoint of 
the midpoint of 
is the projection of 
.
,
,
and 
.
and 
we get 
and 
since 
.
and 
we get 
and 
.
and 
.
we get 
,
combining this result with 
we get 
