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 
.
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
,
.
such that: 
.
Y by
Let 
be a cube with an edge of 
m. Determine the minimum value of the length the segment with the ends on 
and 
and forms an angle of 
with the plane of the face 
.
be a cube with an edge of 
m. Determine the minimum value of the length the segment with the ends on 
and 
and forms an angle of 
with the plane of the face 
.This post has been edited 2 times. Last edited by parmenides51, Mar 18, 2021, 12:34 AM
