[/quote]
,
.
,
on side 
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 Mango247, Mango247, Mango247
Given an acute-angled triangle 
. A straight line parallel to 
intersects sides 
and 
at points 
and 
, respectively. At what location of the points and will the radius of the circle circumscribed about the triangle 
be the smallest?
(I. Sharygin)
. A straight line parallel to 
intersects sides 
and 
at points 
and 
, respectively. At what location of the points and will the radius of the circle circumscribed about the triangle 
be the smallest?(I. Sharygin)
