is acute triangle. 
is orthocenter, 
is the midpoint of 
,
is the midpoint of smaller arc 
is on 
such that, 
is perpendicular to 
.
is paralelogram(Synthetic sol needed).
be the circumcircle of 
.
be a point on the side 
intersects the parallel line to 
through 
.
intersects 
.
,
,
,
are concurrent.
be a positive integer.Prove that there is a polynomial 
with integer coefficients so that 
,![$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$](http://latex.artofproblemsolving.com/5/2/0/520813bebb2f297928208f7b7b2eea76d24043ac.png)
.
.
.
is chosen on side 
is 
.
and 
are chosen on sides 
and 
.
.
to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled 
.
is an equilateral triangle with side length 
.
.
Y by kiyoras_2001
Problem. Let 
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?
Or, is there a function such that 
where 
, and the statement is still true when ?.
be the nonnegative integers in 
where 
. Define 
. Prove that 
.Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?Or, is there a function
such that 
where 
, and the statement is still true when ?.This post has been edited 10 times. Last edited by CBMaster, Apr 27, 2025, 5:43 AM
Reason: .
Reason: .
