[/quote]
,
,
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 
and 
less than 
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 
.
satisfy that: 
and natural number 
Y by
Let 
and 
be the centers of a circle inscribed in a non-isosceles acute-angled triangle 
and circumscribed around it, respectively,. Lines 
and 
intersect the circle circumscribed around the triangle at points 
and 
, respectively. It is known that the points 
lie on one circle. Find the angle .
and 
be the centers of a circle inscribed in a non-isosceles acute-angled triangle 
and circumscribed around it, respectively,. Lines 
and 
intersect the circle circumscribed around the triangle at points 
and 
, respectively. It is known that the points 
lie on one circle. Find the angle .
