[/quote]
,
.
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
,
,
.
is parallel to 
,
is the midpoint of segment 
for which the following statement is true: If positive integers 
and 
are such that 
is a multiple of 
and 
is a multiple of 
and
meaning 
is the least common multiple of the numbers 
.
such that
for every positive integer 
be positive integers with 
.
,
,
,
,
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of 
be an acute triangle with circumcenter 
.
,
on side 
.
passing through 
at point 
and 
is mundane if there exist positive integers 
such that 
is a positive integer. Find, with proof, all prime numbers that are not mundane.
that has the following property: Any 
real numbers, not necessarily distinct, that lie in the interval ![$[100,101]$](http://latex.artofproblemsolving.com/8/e/3/8e3fe24e3b899bbf2128eebe08756e1ff963aaa0.png)
,
and 
such that
.
, 
and 
.
.
-
board.
board consisting of 
unit segments. A configuration of several 
is a divisor of 
.
Y by HWenslawski
Let 
be a fixed rectangular coordinate system in the plane.
Each ordered pair of points
from the same plane which are different from O and have coordinates 
and 
respectively is associated with real number 
in such a way that the following conditions are satisfied:
(a) If
, 
and 
then 
.
(b) There exists a polynomial of second degree
such that 
.
(c) There exists such a number
that for every two points for which 
is satisfied 
.
(d) If the points are such that the triangle 
is equilateral with side 
then
.
Prove that
for each ordered pair of points .
be a fixed rectangular coordinate system in the plane.Each ordered pair of points
from the same plane which are different from O and have coordinates 
and 
respectively is associated with real number 
in such a way that the following conditions are satisfied:(a) If
, 
and 
then 
.(b) There exists a polynomial of second degree
such that 
.(c) There exists such a number
that for every two points for which 
is satisfied 
.(d) If the points
are such that the triangle 
is equilateral with side 
then
.Prove that
for each ordered pair of points .
