Found problems: 339
2013 China National Olympiad, 3
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.
2012 Tuymaada Olympiad, 4
Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the [i]weight of a vertex [/i]the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero.
[i]Proposed by W. Tutte[/i]
2015 District Olympiad, 1
[b]a)[/b] Solve the equation $ x^2-x+2\equiv 0\pmod 7. $
[b]b)[/b] Determine the natural numbers $ n\ge 2 $ for which the equation $ x^2-x+2\equiv 0\pmod n $ has an unique solution modulo $ n. $
2014 Romania National Olympiad, 4
Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $
[b]a)[/b] Prove that the order of $ G $ is a power of $ p. $
[b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $
2012-2013 SDML (Middle School), 5
What is the hundreds digit of the sum below?
$$1+12+123+1234+12345+123456+1234567+12345678+123456789$$
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$
2011 Pre-Preparation Course Examination, 2
prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure.
[b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common.
we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.
1995 Brazil National Olympiad, 2
Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.
2021 Science ON all problems, 2
Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\
$\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\
$\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\
$\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true?
\\ \\
[i] (Bogdan Blaga)[/i]
2017 Danube Mathematical Olympiad, 4
Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.
2005 Miklós Schweitzer, 4
Let F be a countable free group and let $F = H_1> H_2> H_3> \cdots$ be a descending chain of finite index subgroups of group F. Suppose that $\cap H_i$ does not contain any nontrivial normal subgroups of F. Prove that there exist $g_i\in F$ for which the conjugated subgroups $H_i^{g_i}$ also form a chain, and $\cap H_i^{g_i}=\{1\}$.
[hide=Note]Nielsen-Schreier Theorem might be useful.[/hide]
2025 District Olympiad, P1
Let $G$ be a group and $A$ a nonempty subset of $G$. Let $AA=\{xy\mid x,y\in A\}$.
[list=a]
[*] Prove that if $G$ is finite, then $AA=A$ if and only if $|A|=|AA|$ and $e\in A$.
[*] Give an example of a group $G$ and a nonempty subset $A$ of $G$ such that $AA\neq A$, $|AA|=|A|$ and $AA$ is a proper subgroup of $G$.
[/list]
[i]Mathematical Gazette - Robert Rogozsan[/i]
1953 Miklós Schweitzer, 8
[b]8.[/b] Does there exist a Euclidean ring which is properly contained in the field $V$ of real numbers, and whose quotient field is $V$? [b](A.21)[/b]
2010 Contests, 4
Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a
sequence of operations of this type?
Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.
2006 District Olympiad, 2
Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.
1985 Traian Lălescu, 2.3
Let $ 0\neq\varrho\in\text{Hom}\left( \mathbb{Z}_4,\mathbb{Z}_2\right) ,$ $ \text{id}\neq\iota\in\text{Aut}\left( \mathbb{Z}_4\right) ,$ $ G:=\left\{ (x,y)\in\mathbb{Z}_4^2\big|x-y\in\ker\varrho\right\} , $ and $ \rho_1,\rho_2, $ the canonic projections of $ G $ into $ \mathbb{Z}_4. $
Prove that there exists an unique $ \nu\in\text{Hom}\left( \mathbb{Z}_4,G\right) $ such that $ \rho_1\circ\nu=\text{id} $ and $ \rho_2\circ\nu =\iota . $ Determine numerically this morphism.
2013 Miklós Schweitzer, 4
Let $A$ be an Abelian group with $n$ elements. Prove that there are two subgroups in $\text{GL}(n,\Bbb{C})$, isomorphic to $S_n$, whose intersection is isomorphic to the automorphism group of $A$.
[i]Proposed by Zoltán Halasi[/i]
1991 Arnold's Trivium, 92
Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.
2014 IMS, 10
Let $V$ be a $n-$dimensional vector space over a field $F$ with a basis $\{e_1,e_2, \cdots ,e_n\}$.Prove that for any $m-$dimensional linear subspace $W$ of $V$, the number of elements of the set $W \cap P$ is less than or equal to $2^m$ where $P=\{\lambda_1e_1 + \lambda_2e_2 + \cdots + \lambda_ne_n : \lambda_i=0,1\}$.
PEN J Problems, 2
Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]
2024 TASIMO, 6
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 $p.$
2021 CIIM, 3
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.
1969 Miklós Schweitzer, 1
Let $ G$ be an infinite group generated by nilpotent normal subgroups. Prove that every maximal Abelian normal subgroup of $ G$ is infinite. (We call an Abelian normal subgroup maximal if it is not contained in another Abelian normal subgroup.)
[i]P. Erdos[/i]
1978 Germany Team Selection Test, 4
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
2010 Iran MO (3rd Round), 3
If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points)
1993 Hungary-Israel Binational, 5
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 $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?