[/quote]
,
be an acute triangle with circumcenter 
.
,
on side 
passing through 
at point 
.
be a triangle and 
points in the same plane such that 
,
,
.
is parallel to 
,
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, Mango247
In 
, the interior sides of which are mirrors, a laser is placed at point 
on side 
. A laser beam exits the point , hits side 
at point 
, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side 
at point 
, then side at point 
, then side again at point 
, then side again at point 
, then side again at point 
, and finally, side again at point 
.
(a) Prove that
.
(b) Prove that such a laser exists if and only if all the angles in are less than 
.
, the interior sides of which are mirrors, a laser is placed at point 
on side 
. A laser beam exits the point , hits side 
at point 
, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side 
at point 
, then side at point 
, then side again at point 
, then side again at point 
, then side again at point 
, and finally, side again at point 
.(a) Prove that
.(b) Prove that such a laser exists if and only if all the angles in
are less than 
.
and 
so 
