AoPS Academy
+1 w
, 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.
be positive integers such that 
and 
. Find the maximum possible value of 
.
, and this is done by finding the minimum value of 
, 
gives the maximum which yields:
i.e., 
, and so we solve to get 
.
of their two-point shots and 
of their three-point shots, which resulted in 54 points. They attempted