AoPS Academy
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 
.
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
&
&
&
are distinct points and 
.Circumcircles of 
and 
intersect at 
and 
. is the centre of spiral similarity that sends 
to
and to
.also it's the spiral similarity that sends to and 
to. and 
are on a ray Called 
and also both of these points are on one side of on ray and also suppose that and 
are on a ray Called 
and also both of these points are on one side of on ray such that 
. and 
intersect at 
and another point called 
. is the centre of spiral similarity that sends and to and , By our assumption we have 
so the dialation is 1 and we have 
.Regarding to this fact that 
is cyclic , 
is the bisector of 
so is intersection point of the bisector and circumcircle of
.
.We prove that all such varying cirles intersect on this line.
be the point on the ray 
such that lies between and and 
.Hence length 
is a constant
be the point on the angular bisector of 
such that 
Since in any triangle the angular bisector and the perpendicular bisector of the opposite side intersect on the circumcircle which is a fixed point lies on the circumcircle of 
. Let 
so that 
. As constructed, obviously, 
. Let from angle bisector of so that 
is cyclic. Obviously 
and 
. From 
, i.e. 
, hence 
is cyclic, done.
and 
Prove that
vary on two fixed rays emanating from point 
is constant. Show that there is a point 
.