In each vertex of a regular $n$-gon $A_1A_2...A_n$ there is a unique pawn. In each step it is allowed: 1. to move all pawns one step in the clockwise direction or 2. to swap the pawns at vertices $A_1$ and $A_2$. Prove that by a finite series of such steps it is possible to swap the pawns at vertices: a) $A_i$ and $A_{i+1}$ for any $ 1 \leq i < n$ while leaving all other pawns in their initial place b) $A_i$ and $A_j$ for any $ 1 \leq i < j \leq n$ leaving all other pawns in their initial place. [i]Proposed by Matija Bucic[/i]

Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point. [i]Vasile Pop[/i]

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are: (1) A player cannot choose an element that has been chosen by either player on any previous turn. (2) A player can only choose an element that commutes with all previously chosen elements. (3) A player who cannot choose an element on his/her turn loses the game. Patniss takes the first turn. Which player has a winning strategy?

$G$ be a group and $H\le G$. Then which of the following are true? $\textbf{(A)}~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H$ $\textbf{(B)}~\exists G,H\text{ and }H\le G\text{ with }H\cong G$ $\textbf{(C)}~\text{All subgroups are normal, then }G\text{ is abelian.}$ $\textbf{(D)}~\text{None of the above}$

Let $G$ be a finite group of order $n$. Define the set \[H=\{x:x\in G\text{ and }x^2=e\},\] where $e$ is the neutral element of $G$. Let $p=|H|$ be the cardinality of $H$. Prove that a) $|H\cap xH|\ge 2p-n$, for any $x\in G$, where $xH=\{xh:h\in H\}$. b) If $p>\frac{3n}{4}$, then $G$ is commutative. c) If $\frac{n}{2}<p\le\frac{3n}{4}$, then $G$ is non-commutative. [i]Marian Andronache[/i]

Let be the set $ G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} $ and a function $ \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} $ having the property that the operation $ *:G^2\longrightarrow G $ defined as $$ (a,b)*(c,d)=(ac,bc+d\varphi (a)) $$ is associative. [b]a)[/b] Show that $ (G,*) $ is a group. [b]b)[/b] Describe $ \varphi , $ knowing that $(G,*) $ is a commutative group. [i]Marius Perianu[/i]

Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$). Prove that each subgroup of $A$ of finite index is isomorphic to $A$.

Let be a semigroup with the property that for any two elements of it $ a,b, $ there is another element $ c $ such that $ axa=b. $ Prove that it's a group.

Let $ A $ be a ring, $ a\in A, $ and let $ n,k\ge 2 $ be two natural numbers such that $ n\vdots\text{char} (A) $ and $ 1+a=a^k. $ Show that the following propositions are true: [b]a)[/b] $ \forall s\in\mathbb{N}\quad \exists p_0,p_1,\ldots ,p_{k-1}\in\mathbb{Z}_{\ge 0}\quad a^s=\sum_{i=0}^{k-1} p_ia^{i} . $ [b]b)[/b] $ \text{ord} (a)\neq\infty . $

Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group. [b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.

[b]a)[/b] Let $ a,b,c $ be three complex numbers. Prove that the element $ \begin{pmatrix} a & a-b & a-b \\ 0 & b & b-c \\ 0 & 0 & c \end{pmatrix} $ has finite order in the multiplicative group of $ 3\times 3 $ complex matrices if and only if $ a,b,c $ have finite orders in the multiplicative group of complex numbers. [b]b)[/b] Prove that a $ 3\times 3 $ real matrix $ M $ has positive determinant if there exists a real number $ \lambda\in\left( 0,\sqrt[3]{4} \right) $ such that $ A^3=\lambda A+I. $ [i]Cristinel Mortici[/i]

6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.

Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]

Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.

Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\ $\textbf{(a)}$ Prove that $G$ is cyclic. \\ $\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?

Let $ G$ be a locally compact solvable group, let $ c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $ f$ and $ g$ on $ G$ satisfy \[ \sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .\] Prove that if $ f$ is a bounded function and \[ \inf_{x \in G} \textrm{Re} f(x) \chi(x) >0\] for some continuous (complex) character $ \chi$ of $ G$, then $ g$ is continuous. [i]L. Szekelyhidi[/i]

[b]3.[/b]Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$. Show that if, for any $k$, the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$, then every $H_k (k=1, \dots , n)$ is of finite index in $G$. [b](A. 15)[/b]

Let $A$ be a subgroup of the symmetric group $S_n$, and $G$ be a normal subgroup of $A$. Show that if $G$ is transitive, then $|A\colon G|\le 5^{n-1}$ (translated by Miklós Maróti)

Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$. Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have \[ n\mid a_i \minus{} b_i \minus{} c_i \] [i]Proposed by Ricky Liu & Zuming Feng, USA[/i]

Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomorphisms from $G_2$ to $H$. Prove that $G_1$ and $G_2$ are Isomorphic.

Prove that a group $G$ in which exactly two elements other than the identity commute with each other is isomorphic to $\mathbb{Z}/3 \mathbb{Z}$ or $S_3.$

Let be an element $ e $ from a group $ (G,\cdot ) , $ and let be the operation $ *:G^2\longrightarrow G $ defined as $ x*y=x\cdot a\cdot y . $ Prove that $ (G,*) $ is a group and is isomorphic with $ (G,\cdot ) . $

We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$. Say the the following sets have the relation $\sim$ or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$ d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$

There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).

Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]