satisfying the following two conditions:
such that, for all indices 
,
.
and for any index ![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of 
.
and 
be two fixed points on the circle 
be a variable point on 
the intersection point of lines 
and 
,
.
be the circumcircle of triangle 
.
be the second intersection point of 
.
intersects 
.
intersects line 
at 
,
intersects 
.
the second intersection point of the circumcircle of triangle 
with 
into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube 
lie on, and the four planes that the tetrahedron 
lie on. How many regions do these ten planes split the space into?
such that 
.
Y by ehuseyinyigit
Triangle 
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at 
, line 
meet 
at 
. Prove that the three lines 
are concurrent.
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at 
, line 
meet 
at 
. Prove that the three lines 
are concurrent.This post has been edited 1 time. Last edited by pokmui9909, Mar 30, 2025, 5:29 AM
lie on angle bisector of 
.
are symmetric about angle bisector of 
where 
and 
meet at 
is cyclic, so simson on 
and 
be 
,
and the midsegment of 
are concyclic, and by angle chasing we get that the point where 
,
,
is parallel to 
,
,
is symmetrical to the line 
,
is the reflection of 
over 
is perpendicular to the bisector of 
over 
