Game on board with gcd and lcm
by Assassino9931, May 2, 2025, 11:49 PM
On the board are written 
positive integers with least common multiple 
and greatest common divisor 
. It is known that is not a perfect square and is not among the initially written numbers. Two players 
and 
play the following game, taking turns alternatingly, with starting first. In a move the player has to write a number which has not been written so far, by taking two distinct integers 
and 
from the board and write LCM
or LCM/. The player who writes or loses. Who has a winning strategy?
positive integers with least common multiple 
and greatest common divisor 
. It is known that is not a perfect square and is not among the initially written numbers. Two players 
and 
play the following game, taking turns alternatingly, with starting first. In a move the player has to write a number which has not been written so far, by taking two distinct integers 
and 
from the board and write LCM
or LCM/. The player who writes or loses. Who has a winning strategy?
be a positive integer. For a set 
of 
denote the number of increasing arithmetic progressions of length at least two all of whose terms are in ![\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]](http://latex.artofproblemsolving.com/c/8/8/c887443e08b9f61e5ce55c3e35fcf2912a6da2a5.png)
.
children, each of whom selects
be a convex polygon in the plane. Let 
be the set of all midpoints 
of segments 
where 
.
elements. Dissect the polygon 
into triangles so that the following hold:
for some pairwise distinct indices 
.
.
be an acute triangle with 
and 
.
be the midpoint of the side 
.
intersects the side 
again at 
and the circumcircle of triangle 
intersects the side 
again at 
.
lies on the perpendicular bisector of the segment 
and 
.
and 
are tangent.
.
prove that![\[ \prod_{i=1}^k F_i(x_{k-i+1},x_{k-i+2},\ldots,x_k) \geq \prod_{i=1}^k F_i(x_i,x_i,\ldots,x_i)\]](http://latex.artofproblemsolving.com/a/4/d/a4d515f1102b31cff69d6823df7a58482aec1171.png)
.
.
for some real
.
,
is acute, 
,
.
be the intersection of the tangent to the circumcircle of 
and the perpendicular bisector of segment 
.
,
.
May 2025
April 2025