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

Tags were heavily modified to better represent problems.

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Found problems: 339

2009 District Olympiad, 2
Tags: algebra , function , superior algebra , polynomial , abstract algebra
Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

2014 Baltic Way, 18
Tags: abstract algebra , symmetry , modular arithmetic , number theory
Let $p$ be a prime number, and let $n$ be a positive integer. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i\in \{0, 1, \ldots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that \[p^n \mid (a_1a_2 + a_3a_4 + 1).\]

2009 Turkey Team Selection Test, 1
Tags: abstract algebra , prime number , polynomial , algebra , number theory
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

2004 Alexandru Myller, 3
Tags: superior algebra , abstract algebra , ring theory , nilpotence
Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.

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 $f\left( {x + y + f(y)} \right) = f(x) + ny$.

2011 Pre-Preparation Course Examination, 2
Tags: topology , abstract algebra
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: $\forall a,b \in G:ab=ba$.

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 $$ H=\left\{ \left.\left( a^n, b\cdot\frac{1-a^n}{1-a} \right)\right| n\in\mathbb{Z} \right\} $$

1978 Miklós Schweitzer, 2
Tags: abstract algebra , superior algebra
For a distributive lattice $ L$, consider the following two statements: (A) Every ideal of $ L$ is the kernel of at least two different homomorphisms. (B) $ L$ contains no maximal ideal. Which one of these statements implies the other? (Every homomorphism $ \varphi$ of $ L$ induces an equivalence relation on $ L$: $ a \sim b$ if and only if $ a \varphi\equal{}b \varphi$. We do not consider two homomorphisms different if they imply the same equivalence relation.) [i]J. Varlet, E. Fried[/i]

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 $\textbf{(A)}~\text{are necessarily finite}$ $\textbf{(B)}~\text{can be infinite}$ $\textbf{(C)}~\text{can be dense in }S^1$ $\textbf{(D)}~\text{None of the above}$

2023 District Olympiad, P4
Tags: abstract algebra , real analysis , function
Consider the functions $f,g,h:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}$ and the binary operation $*:\mathbb{R}_{\geqslant 0}\times \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}$ defined as \[x*y=f(x)+g(y)+h(x)\cdot|x-y|,\]for all $x,y\in\mathbb{R}_{\geqslant 0}$. Suppose that $(\mathbb{R}_{\geqslant 0},*)$ is a commutative monoid. Determine the functions $f,g,h$.

2008 Putnam, A1
Tags: abstract algebra , algebra , function , limit , calculus , integration
Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$

1969 Canada National Olympiad, 7
Tags: abstract algebra , polynomial , quadratic , domain , modular arithmetic , algebra , function
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.

1985 Traian Lălescu, 1.4
Tags: abstract algebra , superior algebra , ring theory
Let $ A $ be a ring in which $ 1\neq 0. $ If $ a,b\in A, $ then the following affirmations are equivalent: $ \text{(i)}\quad aba=a\wedge ba^2b=1 $ $ \text{(ii)}\quad ab=ba=1 $ $ \text{(iii)}\quad \exists !b\in A\quad aba=a $

2010 Iran MO (3rd Round), 2
Tags: domain , function , integration , algebra , number theory , abstract algebra , calculus
$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

2006 Moldova Team Selection Test, 3
Tags: geometry , circumcircle , function , inequalities , abstract algebra , trigonometry
Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that $a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$

2003 Gheorghe Vranceanu, 1
Tags: group theory , abstract algebra
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.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3
Tags: abstract algebra , superior algebra , group theory
Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.

2025 AIME, 15
Tags: abstract algebra
Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

1990 Turkey Team Selection Test, 6
Tags: abstract algebra , number theory , induction
Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.