[/quote]
,
.
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
,
,
.
is parallel to 
,
is the midpoint of segment 
for which the following statement is true: If positive integers 
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 
.
,
on side 
.
passing through 
at point 
and 
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 
is a divisor of 
.
Y by
Let 
be the center of the circumscribed circle of an acute-angled triangle 
. Let the circle with the center 
circumscribed around the triangle 
, intersects 
for second time at the point 
, and the circle with the center 
, circumscribed around the triangle 
intersects for second time at the point 
. The perpendicular bisector of the segment meets 
at the point 
. Prove that the center of the circumcircle of triangle 
lies on if and only if the points , and are collinear.
be the center of the circumscribed circle of an acute-angled triangle 
. Let the circle with the center 
circumscribed around the triangle 
, intersects 
for second time at the point 
, and the circle with the center 
, circumscribed around the triangle 
intersects for second time at the point 
. The perpendicular bisector of the segment meets 
at the point 
. Prove that the center of the circumcircle of triangle 
lies on if and only if the points , and are collinear.This post has been edited 1 time. Last edited by parmenides51, Jul 3, 2021, 7:57 AM
