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 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.
Y by
On the side 
of the right trapezoid 
(
) such a point 
is chosen that 
. The segments 
and 
are the altitudes of the triangle 
, and 
. Let the point 
, which lies inside the triangle 
, be the intersection point of the segments 
and 
. Prove that is the center of the circumcircle of triangle .
of the right trapezoid 
(
) such a point 
is chosen that 
. The segments 
and 
are the altitudes of the triangle 
, and 
. Let the point 
, which lies inside the triangle 
, be the intersection point of the segments 
and 
. Prove that is the center of the circumcircle of triangle .This post has been edited 2 times. Last edited by parmenides51, Jun 29, 2022, 1:00 PM
