[/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 
at 
and the perpendicular from 
at 
.
does not depend on the choice of 
of parallelogram 
a point 
is chosen. The perpendicular from 
and the perpendicular from 
to 
intersect at 
.
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 
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
Let 
and 
be the centers of the excircles of a non-isosceles triangle 
opposite 
and 
respectively. Let 
and 
be the midpoints of the sides 
and 
respectively Let 
and 
be symmetric to and about and respectively.
Prove that the lines
and 
meet on 
.
and 
be the centers of the excircles of a non-isosceles triangle 
opposite 
and 
respectively. Let 
and 
be the midpoints of the sides 
and 
respectively Let 
and 
be symmetric to and about and respectively.Prove that the lines
and 
meet on 
.
and 
,
has coordinate 
:
whence 
which is symmetric in 
as desired.
be the 
-
and 
are colinear.
and 
and remember that 
and 
and 
and here 
,
and 
.
and 
and now we have:
Now since its easy to see that this determinant (next one i'm showing below) its 0 we are done.
we have the desired result thus we are done 
,
,By some angle chase we know 
.
are collinear. To prove that let 
be the reflection of 
we know that 
are collinear. Since 
and 
,
we can easily deduce that 
,
are perspective from a line, they are perspective from a point (By Desargues theorem), So 
are concurrent.
