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 
The line 
intersects 
Let 
be a point on 
Let 
be the intersection of 
and 
Y by
Let 
be the orthocenter of an isosceles triangle 
(
). The point 
lies on the base 
such that 
, and the point 
lies on the side 
such that 
. Ray 
intersects 
at point 
. Prove that 
.
be the orthocenter of an isosceles triangle 
(
). The point 
lies on the base 
such that 
, and the point 
lies on the side 
such that 
. Ray 
intersects 
at point 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Mar 20, 2021, 9:00 PM
