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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 
.
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
be a positive integer and 
be a family of at most distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
is concave, Anyways, here's a bash,
![$$ [BOC]=\frac{\sin 2A}{\sum \sin 2A} , [AOC]=\frac{\sin 2B}{\sum \sin 2A} , [KOB]= \left( \frac{\sin 2C}{\sum \sin 2A} \right) \left( \frac{\tan A}{\tan A+\tan B} \right) \text{ and } [AOK]=\left( \frac{\sin 2C}{\sum \sin 2A} \right) \left( \frac{\tan B}{\tan A+\tan B} \right) $$](http://latex.artofproblemsolving.com/e/8/0/e80dd370636a6e1da4791f0e6f6d78d1bb8adc66.png)
Now, ![$[BOC]+[KOB]=[AOC]+[AOK]$](http://latex.artofproblemsolving.com/5/9/7/597f23648af40794811bcbb9aa0a1e722a9a743b.png)
Hence, ![$[BKOC]=[AKOC]$](http://latex.artofproblemsolving.com/c/d/5/cd59517b51acf2809e9025e266dc959bbc4bfaca.png)
is proved
be an acute traingle with 
.
is a foot of altitude through vertex 
.
is a circumcenter of 
are equal.
and 
; we have Are
=
Area 
Area 
Area 
Area 
hence the result follows 
be a midpoint of 
are both perpendicular to ![$[CKO]=[CKM]$](http://latex.artofproblemsolving.com/f/c/3/fc3255f4339f6ef935291f012c00e5b728b1e02d.png)
T![$\frac{1}{2}[ABC]=[ACM]=[ACK]+[CKM]=[ACK]+[CKO]=[ACOK]$](http://latex.artofproblemsolving.com/b/0/8/b08f776270fb5dd36f40581a22ae5f528ba4f525.png)
.![$[BCOK]=[ABC]-[ACOK]=\frac{1}{2}[ABC]=[ACOK]$](http://latex.artofproblemsolving.com/6/6/8/6689f753a3964077654adea01821266dbb6fa084.png)
. Point