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 
be a circle with center 
.
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
Consider a collection of 
points on the plane, no three of which are collinear, and some set of line segments between them. We say that a subset of these line segments is called a "pairing" if every one of these () points is the endpoint of exactly one of the chosen line segments (in other words, a pairing is a perfect matching).
Show that for every
, there exists such an arrangement of points and line segments (for some value of 
) such that there are exactly 
distinct (but not necessarily disjoint) pairings.
points on the plane, no three of which are collinear, and some set of line segments between them. We say that a subset of these line segments is called a "pairing" if every one of these () points is the endpoint of exactly one of the chosen line segments (in other words, a pairing is a perfect matching).Show that for every
, there exists such an arrangement of points and line segments (for some value of 
) such that there are exactly 
distinct (but not necessarily disjoint) pairings.This post has been edited 1 time. Last edited by somebodyyouusedtoknow, Apr 26, 2025, 11:37 PM
Reason: for every
Reason: for every
