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]

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$, [*]$x+y=0$, [*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Show that every element of $S_{n}$ is a product of $2$-cycles.

Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.

A space station consists of two living modules attached to a central hub on opposite sides of the hub by long corridors of equal length. Each living module contains $N$ astronauts of equal mass. The mass of the space station is negligible compared to the mass of the astronauts, and the size of the central hub and living modules is negligible compared to the length of the corridors. At the beginning of the day, the space station is rotating so that the astronauts feel as if they are in a gravitational field of strength $g$. Two astronauts, one from each module, climb into the central hub, and the remaining astronauts now feel a gravitational field of strength $g'$ . What is the ratio $g'/g$ in terms of $N$?[asy] import roundedpath; size(300); path a = roundedpath((0,-0.3)--(4,-0.3)--(4,-1)--(5,-1)--(5,0),0.1); draw(scale(+1,-1)*a); draw(scale(+1,+1)*a); draw(scale(-1,-1)*a); draw(scale(-1,+1)*a); filldraw(circle((0,0),1),white,black); filldraw(box((-2,-0.27),(2,0.27)),white,white); draw(arc((0,0),1.5,+35,+150),dashed,Arrow); draw(arc((0,0),1.5,-150,-35),dashed,Arrow);[/asy] $ \textbf{(A)}\ 2N/(N-1) $ $ \textbf{(B)}\ N/(N-1) $ $ \textbf{(C)}\ \sqrt{(N-1)/N} $ $ \textbf{(D)}\ \sqrt{N/(N-1)} $ $ \textbf{(E)}\ \text{none of the above} $

Consider the additive group $\mathbb{Z}^{2}$. Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$. Let $H_{xy}$ be the smallest subgroup containing $(0,x)$ and $(1,y)$. Find some pair $(x,y)$ with $x>0$ such that $H=H_{xy}$.

Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$). 1) If $k=2n+1$, prove that there exists a person who knows all others. 2) If $k=2n+2$, give an example of such a group in which no-one knows all others.

Prove that any permutation group of an order equal to a power of $ 2 $ contains a commutative subgroup whose order is the square of the exponent of the order of the group.

Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums \[a_{k_1}\] \[a_{k_1}+a_{k_2}\] \[a_{k_1}+a_{k_2}+a_{k_3}\] \[\vdots\] \[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\] have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.

Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$). a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$; b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$. [i]Calin Popescu[/i]

What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?

Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that: $$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$ [i]Proposed by Ovidiu Pop[/i]

Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that \[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\] for every positive integer $k$. [i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?

Show that the equation $ (x+y)^{-1}=x^{-1}+y^{-1} $ has a solution in the field of integers modulo $ p $ if and only if $ p $ is a prime congruent to $ 1 $ modulo $ 3. $ [i]Mihai Piticari[/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 $ \mathcal G$ be the set of all finite groups with at least two elements. a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$. b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.

Let $m,n$ and $N$ be positive integers and $\mathbb{Z}_{N}=\{0,1,\dots,N-1\}$ a set of residues modulo $N$. Consider a table $m\times n$ such that each one of the $mn$ cells has an element of $\mathbb{Z}_{N}$. A [i]move[/i] is choose an element $g\in \mathbb{Z}_{N}$, a cell in the table and add $+g$ to the elements in the same row/column of the chosen cell(the sum is modulo $N$). Prove that if $N$ is coprime with $m-1,n-1,m+n-1$ then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.

Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$. i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian. ii) Give an example of a non-abelian group with $ G$'s property from the enounce.

Let be a group with $ 10 $ elements for which there exist two non-identity elements, $ a,b, $ having the property that $ a^2 $ and $ b^2 $ are the identity. Show that this group is not commutative.

Let $P(x)$ be polynomial with integer coefficients. Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$