AoPS Academy
be an acute triangle with circumcenter 
and orthocenter 
. Let 
denote the circumcircle of triangle , and 
the midpoint of 
. The tangents to at 
and 
, and the line through perpendicular to line 
, determine a triangle whose circumcircle we denote by 
. Define 
and 
similarly., and are concurrent on line .
such that for all positive integers 
and 
,![\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]](http://latex.artofproblemsolving.com/8/b/9/8b90a9326c0fa448a5a574809c184f43513f768f.png)
Jaedon Whyte
, and an odd prime 
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
, let 
be the least prime that does not divide 
, and define 
to be the product of all primes less than . In particular, 
For having 
, define 
. Consider the sequence 
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
. Find all such that 
.
be a triangle with incenter 
and excenters 
, 
, and 
opposite 
, , and , respectively. Let 
be an arbitrary point on the circumcircle of 
that does not lie on any of the lines 
, 
, or 
. Suppose the circumcircles of 
and 
intersect at two distinct points and 
. If 
is the intersection of lines 
and , prove that 
. for which 
is a perfect square.
be a cube with length side 
and 
,
, 
. Find minimum perimeter of 
.
be a cube with length side and ,, . Find minimum perimeter of .
,
are the midpoints.