[/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 GA34-261
3. [Russia 1961]
Real numbers are written in an
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
Real numbers are written in an
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.This post has been edited 2 times. Last edited by CatalanThinker, May 28, 2025, 5:43 AM
