[/quote]
,
be an acute triangle with circumcenter 
.
.
passing through 
at point 
.
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, Mango247
Let 
be a cube and 
a variable point on the side ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. The perpendicular plane on 
which passes through intersects the line 
in 
. Let 
and 
be the midpoints of the segments 
and 
respectively.
a) Prove that the lines
and 
are perpendicular if and only if is the midpoint of .
b) Find the minimal value of the angle between the lines and .
be a cube and 
a variable point on the side ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. The perpendicular plane on 
which passes through intersects the line 
in 
. Let 
and 
be the midpoints of the segments 
and 
respectively.a) Prove that the lines
and 
are perpendicular if and only if is the midpoint of .b) Find the minimal value of the angle between the lines
and .
