AoPS Academy
, the numbers 
are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is 
after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out., determine which player (if any) has a winning strategy
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
satisfies 
, and its incenter is 
. 
is the foot from to ，and 
is a point on segment such that 
. Line 
meets 
at point 
, and the orthocenter of 
is .
is tangent to 
.
be the reflection of 
over 
. We claim that is the desired tangency point. First note that by simple angle chase 
is cyclic. lies on 
.
over lies on 
, which implies that the reflection of over lies on as desired. lies on 
.
and 
we have 
. is cyclic. Now note that![\[\measuredangle DI'C=\measuredangle CID =\measuredangle BEC=\measuredangle BFC=\measuredangle DFC\]](http://latex.artofproblemsolving.com/2/f/5/2f58a17403d926079fb39c1cd298cc4f382e0d25.png)
so 
is cyclic.![\[\angle DI'E = \angle EID = 90^\circ - \angle DEI = 90^\circ -\frac{\angle B}{2}=180^\circ - \angle AIC = \angle IAC+\angle ACI =\angle I'AC + \angle I'CA\]](http://latex.artofproblemsolving.com/a/d/e/ade4b498e12c001340fcb4941334172441534b99.png)
So the circles 
and 
are tangent, and we are done by claim 1 and 2.
