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 Mango247, Mango247, Mango247
Consider a positive integer 
and two points 
and 
in a plane. Starting from point , rays and starting from point , rays are drawn so that all of them are on the same half-plane defined by the line 
and that the angles formed by the 
rays with the segment are all acute. Define circles passing through points , and each meeting point between the rays. What is the smallest number of distinct circles that can be defined by this construction?
and two points 
and 
in a plane. Starting from point , rays and starting from point , rays are drawn so that all of them are on the same half-plane defined by the line 
and that the angles formed by the 
rays with the segment are all acute. Define circles passing through points , and each meeting point between the rays. What is the smallest number of distinct circles that can be defined by this construction?This post has been edited 2 times. Last edited by parmenides51, Sep 15, 2022, 3:57 PM
