Found problems: 6
Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$).
Proposed by Heidar Shushtari
We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$
We say that a prime $p$ is $\textit{philé}$ if there is a polynomial $P$ of non-negative integer coefficients smaller than $p$ and with degree $3$, that is, $P(x) = ax^3 + bx^2 + cx + d$ where $a, b, c, d < p$, such that $$\{P(n) | 1 \leq n \leq p\}$$ is a complete residue system modulo $p$. Find all $\textit{philé}$ primes.
Note: A set $A$ is a complete residue system modulo $p$ if for every integer $k$, with $0 \leq k \leq p - 1$, there exists an element $a \in A$ such that $$p | a-k.$$
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]
A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds:
For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$.
(Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$)
Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]