[/quote]
,
be an acute triangle with circumcenter 
.
,
on side 
.
passing through 
at point 
.
and 
be a triangle and 
points in the same plane such that 
,
,
.
is parallel to 
,
is the midpoint of segment 
,
and
meaning 
is the least common multiple of the numbers 
.
such that
for every positive integer 
such that 
for any relatively prime natural numbers 
.
is described on a circle 
,
be 
respectively. A circle with a center 
,
.
,
.
,
and lines 
,
and lines 
are concurrent.
is a divisor of 
.
,
and 
.
is prime, 
with 
and 
.
and 
are such that 
is a multiple of 
and 
is a multiple of 
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 
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 
.
.
Y by Adventure10
Given a scalene triangle 
. 
are points lie on the rays 
such that 
. Now, for an arbitrary point 
in the plane. Let 
be the reflection point of w.r.t 
. The intersections of 
and 
is 
and the intersections of 
and 
is 
. Suppose that 
are circumcenters of 
Show that
1.
are colinear
2.
. 
are points lie on the rays 
such that 
. Now, for an arbitrary point 
in the plane. Let 
be the reflection point of w.r.t 
. The intersections of 
and 
is 
and the intersections of 
and 
is 
. Suppose that 
are circumcenters of 
Show that1.
are colinear2.
means that 
lies on the radical axis of 
,
collinear, and similarly 
collinear with 
.
.
symmetric about 
(1) and 
(2). Now invert about 
,
and maps 
.
and (2) becomes 
.
collinear, and (2) implies 
.
