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 
Let 
be the midpoint of 
The line 
intersects 
Let 
be a point on 
Let 
be the intersection of 
and 
Y by Mango247, Mango247, Mango247
On sides 
and 
of an equilateral triangle 
with side 
, points 
and 
are taken, respectively, such that the segment 
touches the circle inscribed in the triangle and its length is 
. Find the area of the triangle 
.
and 
of an equilateral triangle 
with side 
, points 
and 
are taken, respectively, such that the segment 
touches the circle inscribed in the triangle and its length is 
. Find the area of the triangle 
.
![\[
AP+AQ=2AB-CQ-BP=2AB-AB-PQ=AB-PQ=10-4=6
\]](http://latex.artofproblemsolving.com/3/4/7/347bc101441bce8fbc693313e57079c58778a679.png)
So 
.
