and 
Prove that
such that 
and 
for all reals 
.
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 
.
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
Y by Adventure10, Mango247
On a circle of radius 
, the distinct points 
, 
, 
, 
, and 
lie in this order, satisfying 
. Show that the triangle with vertices lying in the centroids of the triangles 
, 
, and 
is obtuse.
Proposed by Tomáš Jurík, Slovakia
, the distinct points 
, 
, 
, 
, and 
lie in this order, satisfying 
. Show that the triangle with vertices lying in the centroids of the triangles 
, 
, and 
is obtuse.Proposed by Tomáš Jurík, Slovakia
This post has been edited 1 time. Last edited by Radar, Jun 19, 2015, 8:26 PM
,
,
the centroids of 
,
,
,
.
and 
.
.
,
then 
,
