[/quote]
,
such that 
is a divisor of 
.
,
and 
.
is prime, 
with 
and 
.
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
,
,
.
is parallel to 
,
is the midpoint of segment 
and 
are such that 
is a multiple of 
and 
is a multiple of 
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 
be an acute triangle with circumcenter 
,
passing through 
at point 
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 
.
.
-
board.
board consisting of 
unit segments. A configuration of several 
Y by jhu08, Rounak_iitr
Let 
be a quadrilateral inscribed in a circle with center 
. Lines 
, 
meet at 
and lines 
, 
meet at 
. Let 
be the second intersection point of the circumscribed circles of the triangles 
and 
.
a) Prove that
and are collinear.
b) Let
and 
be the circumcenters of the triangles and , respectively. Prove that the points 
are concyclic.
be a quadrilateral inscribed in a circle with center 
. Lines 
, 
meet at 
and lines 
, 
meet at 
. Let 
be the second intersection point of the circumscribed circles of the triangles 
and 
.a) Prove that
and are collinear.b) Let
and 
be the circumcenters of the triangles and , respectively. Prove that the points 
are concyclic.
