[/quote]
,
be an odd prime number. Let 
be the sequence defined as follows: 
,
,
is the smallest integer greater than 
such that in 
there are no arithmetic progressions of length 
.
with 
,
positive integers 
such that for every index 
,
,
is a multiple of 
.
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
.
is a prime number, prove that the number 
is composite.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
be real numbers such that 
Prove that 
Where 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
Prove that 
,
,
,
can be expressed by a sum of positive integers 
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
equally spaced points on a circular track 
of circumference 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
.
,
,
,
to 
,
of 
is a diameter of 
Y by
Let 
be a given positive integer. Consider a permutation of 
, denoted as 
. For a section 
, we call it "extreme" if 
and 
are the maximum and minimum value of that section. We say a permutation is "super balanced" if there isn't an "extreme" section with a length at least 
. For example, 
is "super balanced", but 
isn't. Please answer the following questions:
1. How many "super balanced" permutations are there?
2. For each integer
. How many "super balanced" permutations are there such that 
?
Proposed by ltf0501
be a given positive integer. Consider a permutation of 
, denoted as 
. For a section 
, we call it "extreme" if 
and 
are the maximum and minimum value of that section. We say a permutation is "super balanced" if there isn't an "extreme" section with a length at least 
. For example, 
is "super balanced", but 
isn't. Please answer the following questions:1. How many "super balanced" permutations are there?
2. For each integer
. How many "super balanced" permutations are there such that 
?Proposed by ltf0501
This post has been edited 1 time. Last edited by CrazyInMath, Sep 6, 2022, 10:28 AM
somewhere, and each number can be on the left or right of 
.
by utilizing Vandermonde’s for 
and inspection for 
.
