[/quote]
,
such that
for all 
.
such that 
.
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
be a triangle with circumcircle 
and circumcenter 
.
and 
represent the 
-
the circle tangent to 
,
,
.
with line 
,
be the intersection point of 
and 
as the point on segment 
such that 
.
intersects 
.
be the touch point of the 
be the antipode of 
be the intersection of 
and 
.
is the radical axis of 
and 
be points on circle 
,
,
tangent to 
be any point on arc 
,
,
,
to circles 
.
,
,
,
be points on circle 
equal parts. Construct 
tangent to 
,
,
to circles 
of 
(where 
)
denote the set of positive integers
such that for all 
it holds that 
Note: consider 
and 
and 
Prove that
Where 
Y by OronSH, Kingsbane2139
Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
This post has been edited 1 time. Last edited by mathbetter, Mar 20, 2025, 11:21 AM
