  In this paper we investigate some new problems in additive combinatorics. Our
problems mainly involve permutations (or circular permutations) $n$ distinct
numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with
adjacent sums $a_i+a_{i+1}$ (or differences $a_i-a_{i+1}$) pairwise distinct.
For an odd prime power $q=2n+1>13$ with $q\not=25$, we show that there is a
circular permutation $(a_1,\ldots,a_n)$ of the elements of $S=\{a^2:\
a\in\mathbb F_q\setminus\{0\}\}$ such that
$\{a_1+a_2,\ldots,a_{n-1}+a_n,a_n+a_1\}=S$, where $\mathbb F_q$ denotes the
field of order $q$. For any finite subset $A$ of an additive torsion-free
abelian group $G$ with $|A|=n>3$, we prove that there is a numbering
$a_1,\ldots,a_n$ of the elements of $A$ such that $$a_1+2a_2,\ a_2+2a_3,\
\ldots,\ a_{n-1}+2a_n,\ a_n+2a_1$$ are pairwise distinct. We also pose 30 open
conjectures for further research.
