[/quote]
,
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)
,
,
be a triangle in which 
.
and 
be the bisectors of 
and 
with 
on 
and 
on 
.
be the reflection of 
in line 
.
.
good if for any distinct positive integers 
,
Call a positive integer 
to be 
-
.
.
,
and 
.
.
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
be a real number. Determine all functions 
such that ![\[f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)\]](http://latex.artofproblemsolving.com/c/2/a/c2a12ab93b2e38c3ef6e8402edb2858a3d8d12cd.png)
holds for all 
be a positive integer. Given is a subset 
with 
elements. Prove that there exist three elements 
from 
.
be positive integers with product 
where 
of integers is rootiful if, for any positive integer 
,
are also in 
for positive integers 
.
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.
