[/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 
.
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 Mango247
Two circles with no common points are given and their common external tangents are drawn. One of them touches the first circle at point 
and the other touches the second at point 
. The line 
again intersects the first circle at point 
and the second at point 
. Prove that 
.
and the other touches the second at point 
. The line 
again intersects the first circle at point 
and the second at point 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Oct 4, 2020, 10:04 AM
Reason: typo, replaced // with =
Reason: typo, replaced // with =
)
.![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
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