[/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 
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
Consider a semicircle of center 
and diameter ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, and let 
be an arbitrary point on the segment 
. The perpendicular to the line 
through intersects the semicircle in 
. A circle centered in 
is tangent to the arc 
in 
and to the segments and ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
in 
and 
, respectively. Prove that the triangle 
is isosceles.
and diameter ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, and let 
be an arbitrary point on the segment 
. The perpendicular to the line 
through intersects the semicircle in 
. A circle centered in 
is tangent to the arc 
in 
and to the segments and ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
in 
and 
, respectively. Prove that the triangle 
is isosceles.This post has been edited 1 time. Last edited by parmenides51, May 29, 2020, 6:52 AM
are collinear or notice that 
are collinear and 
since 
(this is just simulating the homothety). So by Power of point we have 
(EFBC cyclic and ADC similar to ABD) so we're done.
