[/quote]
,
.
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
,
,
.
is parallel to 
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 
.
,
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
Let 
be a triangle and let 
be a point inside it (not on the sides). Let 
be the the other intersection point of the line 
and the circle circumscribed to , let 
be the midpoint of side 
, and let 
be the symmetric of wrt to . We define in a way similar points 
and 
(starting from 
).
(a) Determine (if any) all points for which 
are not three points distinct.
(b) In cases where are three distinct points, let K be the circumcenter of the triangle 
. Prove that, as varies, the midpoint of the segment 
remains fixed.
be a triangle and let 
be a point inside it (not on the sides). Let 
be the the other intersection point of the line 
and the circle circumscribed to , let 
be the midpoint of side 
, and let 
be the symmetric of wrt to . We define in a way similar points 
and 
(starting from 
).(a) Determine (if any) all points
for which 
are not three points distinct.(b) In cases where
are three distinct points, let K be the circumcenter of the triangle 
. Prove that, as varies, the midpoint of the segment 
remains fixed.
.
the orthocenter 
.
.
