[/quote]
,
be the incenter, centroid, and circumcenter of the nonisosceles triangle 
.
if and only if 
.
and circumcircle 
.
and 
be the second intersection points of 
and 
,
meets 
at a point 
,
at a point 
.
be the intersection point of the line through 
and the line through 
.
and 
meet at a point 
.
and 
are either parallel or concurrent.
be a triangle with circumcentre 
.
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
.
be the circle passing through 
.
be the midpoint of 
be the midpoint of 
.
.
and 
are collinear.
(where 
are on the sides of the triangle), and their intersection point is 
lies on the circumcircle of 
,
is right. The circumcircle of 
a second time at the point 
.
lies on line 
and 
.
and 
the circumcentre and orthocentre, respectively, of the triangle 
Suppose that the circumcircle of the triangle 
intersects the line 
different from 
intersects the line 
Prove that the circumcentre of the triangle 
lies on the line 
.
be an arbitrary point on the side 
of triangle 
.
and 
be the intersections of the lines through 
and 
,
be the intersection point of the circumcircle of 
with the circumcircle of 
be the midpoint of 
be the intersection point of line 
with the circumcircle of 
are collinear.
of integers such that ![\[1+2^{x}+2^{2x+1}= y^{2}.\]](http://latex.artofproblemsolving.com/9/c/8/9c81ffa59a374578a1dba99ad772902ea17ebadb.png)
![$f\colon \mathbb{Z}^2 \to [0, 1]$](http://latex.artofproblemsolving.com/3/2/e/32e14eccb91bbe356013e2a6e5a37a2b54223e08.png)
such that for any integers 
and 
,![\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]](http://latex.artofproblemsolving.com/9/a/0/9a0933a0d4c566c8201d1a26053bc992a21d4d45.png)
,
,
.
and 
with 
.
good if for any distinct positive integers 
,
Call a positive integer 
to be 
-
.
.
Y by
5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
