and 
Prove that
is given, find the largest real number 
such that there exists a geometric sequence 
with common ratio 
for all positive integers 
.
denotes the distance from the real number 
to the nearest integer.
,
.
.
is your favourite function, 
is your mother's favourite function, and 
is your father's favourite function. It was discovered that for any reals 
,
Prove that you are boring.
Y by
Define sequence 
as follows: 
, 
, for 
, 
is the least positive integer such that there is exactly one way to represent as 
where 
. Question: does there exists a constant 
, such that for any positive integer 
, ![\[\min_{1\le k\le n}\frac{a_{k+1}}{a_k}\le 1+\frac{c\ln n}n\text{ }?\]](//latex.artofproblemsolving.com/a/3/2/a32c044b4bb701674aa885f1b0737f2a4eaa2d85.png)
as follows: 
, 
, for 
, 
is the least positive integer such that there is exactly one way to represent as 
where 
. Question: does there exists a constant 
, such that for any positive integer 
, ![\[\min_{1\le k\le n}\frac{a_{k+1}}{a_k}\le 1+\frac{c\ln n}n\text{ }?\]](http://latex.artofproblemsolving.com/a/3/2/a32c044b4bb701674aa885f1b0737f2a4eaa2d85.png)
This post has been edited 1 time. Last edited by EthanWYX2009, Mar 26, 2025, 11:49 AM
