be an acute triangle with 
,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
and 
.
.
be the point on 
.
by 
,
for every positive integer 
.
leave distinct remainders when divided by 
.
,
from all multiples of 
-
.
which are simultaneously 
-
-
-
.
satisfying 
and![\[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.\]](http://latex.artofproblemsolving.com/0/4/4/0446861fb2e63782803c4f182a529b79e213fdfd.png)
satisfies 
,
.
and 
intersect at point 
.
Y by
Let 
be an integer. Two players, Alice and Bob, play the following game on the complete graph 
: They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.
Prove that there exists a positive number
, Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding
![\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]](//latex.artofproblemsolving.com/a/9/c/a9cca00e87c643af32eb75a2501b59060f3504fb.png)
be an integer. Two players, Alice and Bob, play the following game on the complete graph 
: They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.Prove that there exists a positive number
, Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding![\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]](http://latex.artofproblemsolving.com/a/9/c/a9cca00e87c643af32eb75a2501b59060f3504fb.png)
This post has been edited 1 time. Last edited by EthanWYX2009, Mar 29, 2025, 2:56 PM
.
