[/quote]
,
and 
Prove that
such that 
and 
for all reals 
.
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
be a positive integer and 
be a family of at most 
with the following property: we can consider 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
such that the equation 
has an integer solution 
.
such tha
(Adit Aggarwal, India.)
and some point 
on 
.
intersects the circumcircle of 
at 
intersects the circumcircle of 
at 
does not depend on the choice of 
of parallelogram 
a point 
is chosen. The perpendicular from 
to 
and the perpendicular from 
to 
intersect at 
is the midpoint of 
.
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
is prime.
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
is an integer.
?
Y by Adventure10
A solid right circular cylinder with height 
and base-radius 
has a solid hemisphere of radius resting upon it. The center of the hemisphere 
is on the axis of the cylinder. Let 
be any point on the surface of the hemisphere and 
the point on the base circle of the cylinder that is furthest from (measuring along the surface of the combined solid). A string is stretched over the surface from to so as to be as short as possible. Show that if the string is not in a plane, the straight line 
when produced cuts the curved surface of the cylinder.
and base-radius 
has a solid hemisphere of radius resting upon it. The center of the hemisphere 
is on the axis of the cylinder. Let 
be any point on the surface of the hemisphere and 
the point on the base circle of the cylinder that is furthest from (measuring along the surface of the combined solid). A string is stretched over the surface from to so as to be as short as possible. Show that if the string is not in a plane, the straight line 
when produced cuts the curved surface of the cylinder.
