and 
Prove that
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 
.
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects 
Let 
be a point on 
Let 
be the intersection of 
and 
Y by
Given a triangle 
and a point 
. The lines 
,
,
hit the opposite side of the triangle at 
respectively. On the exterior of , we construct three pairs of similar triangles: 
,
on 
,
, 
,
on 
,
, and 
,
on 
, 
. The lines 
,
,
intersect each other form a triangle 
. Prove that 
,
,
are concurrent.
and a point 
. The lines 
,
,
hit the opposite side of the triangle at 
respectively. On the exterior of , we construct three pairs of similar triangles: 
,
on 
,
, 
,
on 
,
, and 
,
on 
, 
. The lines 
,
,
intersect each other form a triangle 
. Prove that 
,
,
are concurrent.
