This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 250

2003 Romania National Olympiad, 4
Tags: limit , ring theory , group theory , abstract algebra
$ i(L) $ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $ L $ such that the set of elements of $ L, $ along with these additive and multiplicative operations, form a ring. Prove that [b]a)[/b] $ i\left( \mathbb{Z}_{12} \right) =4. $ [b]b)[/b] $ i(A\times B)\ge i(A)i(B) , $ for any two finite commutative groups $ B $ and $ A. $ [b]c)[/b] there exist two sequences $ \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} $ of finite commutative groups such that $$ \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 $$ and $$ \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty. $$ [i]Barbu Berceanu[/i]

2012 Grigore Moisil Intercounty, 1
Tags: group theory
[b]a)[/b] Find the group $ H $ that is isomorphic with the multiplicative group of positive real numbers, having an isomorphism $$ \iota :(0,\infty )\longrightarrow H,\quad\iota (x)=\frac{x-1}{x+1} . $$ [b]b)[/b] Calculate the $ 2012\text{-th} $ power of an arbitrary element of $ H. $

2021 Science ON grade XII, 4
Tags: group theory , superior algebra
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?

1993 Hungary-Israel Binational, 1
Tags: superior algebra , abstract algebra , group theory
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. Suppose $k \geq 2$ is an integer such that for all $x, y \in G$ and $i \in \{k-1, k, k+1\}$ the relation $(xy)^{i}= x^{i}y^{i}$ holds. Show that $G$ is Abelian.

1987 Traian Lălescu, 1.1
Tags: superior algebra , group theory
Describe all groups $ G $ which have the property that: $$ (\forall H\le G)(\forall x,y\in G)(xy\in H\implies (x,y\in H\vee xy=1)) $$

2025 District Olympiad, P1
Tags: group theory , abstract algebra
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]

2008 IMC, 4
Tags: abstract algebra , matrix , inequalities , linear algebra , analytic geometry , group theory , geometry
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

2014 IMS, 7
Tags: superior algebra , group theory , abstract algebra
Let $G$ be a finite group such that for every two subgroups of it like $H$ and $K$, $H \cong K$ or $H \subseteq K$ or $K \subseteq H$. Prove that we can produce each subgroup of $G$ with 2 elements at most.

2004 Gheorghe Vranceanu, 1
Tags: superior algebra , group theory , abstract algebra
Let $(G,\cdot)$ be a group, and let $H_1,H_2$ be proper subgroups s.t. $H_1\cap H_2=\{e\}$, where $e$ is the identity element of $G$. They also have the following properties: [b]i)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_1\setminus\{e\}\Rightarrow xy\in H_2$ [b]ii)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_2\setminus\{e\}\Rightarrow xy\in H_1$ Prove that: [b]a)[/b] $|H_1|=|H_2|$ [b]b)[/b] $|G|=|H_1|\cdot |H_2|$

2012 France Team Selection Test, 1
Tags: group theory , graph theory , abstract algebra , combinatorics
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.

2019 Romania National Olympiad, 4
Tags: matrix , prime number , permutation , linear algebra , group theory
Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

2024 Miklos Schweitzer, 7
Tags: abstract algebra , group theory
Is it true that if a subgroup $G \leq \text{Sym}(\mathbb{N})$ is $n$-transitive for every positive integer $n$, then every group automorphism of $G$ extends to a group automorphism of $\text{Sym}(\mathbb{N})$?

2000 District Olympiad (Hunedoara), 1
Tags: group , binary operation , semigroups , superior algebra , group theory
Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $ [b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication. [b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups? [b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $

1973 Miklós Schweitzer, 1
Tags: abstract algebra , modular arithmetic , galois theory , superior algebra , matrix , group theory , linear algebra
We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$. [i]J. Erdos[/i]

1967 Miklós Schweitzer, 3
Tags: superior algebra , abstract algebra , group theory
Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup. [i]B. Csakany[/i]

PEN H Problems, 49
Tags: abstract algebra , diophantine equation , search , group theory
Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

2005 MOP Homework, 1
Tags: abstract algebra , number theory , group theory
We call a natural number 3-partite if the set of its divisors can be partitioned into 3 subsets each with the same sum. Show that there exist infinitely many 3-partite numbers.

2012 China Team Selection Test, 3
Tags: abstract algebra , group theory , algebra , floor function , function , induction
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

2012 Bogdan Stan, 1
Tags: group theory
Find the number of pairs of elements, from a group of order $ 2011, $ such that the square of the first element of the pair is equal to the cube of the second element. [i]Teodor Radu[/i]

1957 Miklós Schweitzer, 10
Tags: abstract algebra , group theory
[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]

2005 Grigore Moisil Urziceni, 3
Tags: abstract algebra , group theory
Define the operation $ (a,b)\circ (c,d) =(ac,ad+b). $ [b]a)[/b] Prove that $ \left( \mathbb{Q}\setminus\{ 0\}\times\mathbb{Q} ,\circ \right) $ is a group. [b]b)[/b] Let $ H $ be an infinite subgroup of $ \left( \mathbb{Q}\setminus\{ 0\}\times\mathbb{Q} ,\circ \right) $ that is cyclic and doesn't contain any element of the form $ (1,q) , $ where $ q $ is a nonzero rational. Show that there exist two rational numbers $ a,b $ such that $$ H=\left\{ \left.\left( a^n, b\cdot\frac{1-a^n}{1-a} \right)\right| n\in\mathbb{Z} \right\} $$

2011 Iran MO (3rd Round), 1
Tags: 3d geometry , dodecahedron , rotation , geometry , group theory , geometric transformation
A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron. [b]a)[/b] prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before. [b]b)[/b] prove that the number four in previous part can't be replaced with three. [i]proposed by Kasra Alishahi[/i]

2013 Princeton University Math Competition, 2
Tags: group theory , distinguishability , rotation , counting
How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.

2020 Jozsef Wildt International Math Competition, W17
Tags: abstract algebra , field , group theory , field theory
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]

2019 LIMIT Category C, Problem 5
Tags: abstract algebra , group theory
Let $G=(S^1,\cdot)$ be a group. Then its nontrivial subgroups $\textbf{(A)}~\text{are necessarily finite}$ $\textbf{(B)}~\text{can be infinite}$ $\textbf{(C)}~\text{can be dense in }S^1$ $\textbf{(D)}~\text{None of the above}$