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: 84

2024 District Olympiad, P3
Tags: number theory , Ring Theory
Let $k$ be a positive integer. A ring $(A,+,\cdot)$ has property $P_k$ if for any $a,b\in A$ there exists $c\in A$ such that $a^k=b^k+c^k.$[list=a] [*]Give an example of a finite ring $(A,+,\cdot)$ which [i]does not[/i] have $P_k$ for any $k\geqslant 2.$ [*]Let $n\geqslant 3$ be an integer and $M_n=\{m\in\mathbb{N}:(\mathbb{Z}_n,+,\cdot)\text{ has }P_m\}.$ Prove that all the elements of $M_n$ are odd integers and that $(M_n,\cdot)$ is a monoid. [/list]

2008 Romania National Olympiad, 3
Tags: Ring Theory , linear algebra , matrix , number theory , least common multiple , search , group theory
Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that a) $ 0$ is the only nilpotent element of $ A$; b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.

1976 Miklós Schweitzer, 4
Tags: calculus , integration , algebra , function , domain , polynomial , Ring Theory
Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

2001 Miklós Schweitzer, 3
Tags: Miklos Schweitzer , abstract algebra , Ring Theory , Matrices
How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?

2001 Miklós Schweitzer, 4
Tags: Miklos Schweitzer , abstract algebra , Units , Ring Theory
Find the units of $R=\mathbb Z[t][\sqrt{t^2-1}]$.

2017 District Olympiad, 4
Tags: superior algebra , idempotence , Ring Theory
Let $ A $ be a ring that is not a division ring, and such that any non-unit of it is idempotent. Show that: [b]a)[/b] $ \left( U(A) +A\setminus\left( U(A)\cup \{ 0\} \right) \right)\cap U(A) =\emptyset . $ [b]b)[/b] Every element of $ A $ is idempotent.

2012 Centers of Excellency of Suceava, 2
Tags: abstract algebra , matrix ring , Ring Theory
Show that $$ \left\{ X\in\mathcal{M}_2\left( \mathbb{Z}_3 \right)\left| \begin{pmatrix} 1&1\\2&2 \end{pmatrix} X\begin{pmatrix} 1&2\\2&1 \end{pmatrix} =0 \right. \right\} $$ is a multiplicative ring. [i]Cătălin Țigăeru[/i]

2014 Romania National Olympiad, 1
Tags: function , Ring Theory
For a ring $ A, $ and an element $ a $ of it, define $ s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$ [b]a)[/b] Prove that if $ A $ is finite, then $ s_a $ is injective if and only if $ d_a $ is injective. [b]b)[/b] Give example of a ring which has an element $ b $ for which $ s_b $ is injective and $ d_b $ is not, or, conversely, $ s_b $ is not injective, but $ d_b $ is.

2005 Romania National Olympiad, 4
Tags: induction , topology , Ring Theory , superior algebra , superior algebra unsolved
Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let \[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \] Prove that the following statements are equivalent: a) $A$ is a field; b) $M$ is not empty and the smallest element in $M$ is $2^n+1$. [i]Marian Andronache[/i]

2010 N.N. Mihăileanu Individual, 4
Tags: modular arithmetic , Ring Theory , abstract algebra , Totient
If $ p $ is an odd prime, then the following characterization holds. $$ 2^{p-1}\equiv 1\pmod{p^2}\iff \sum_{2=q}^{(p-1)/2} q^{p-2}\equiv -1\pmod p $$ [i]Marius Cavachi[/i]

Gheorghe Țițeica 2024, P3
Tags: Ring Theory , abstract algebra
Determine all commutative rings $R$ with at least four elements that are not fields, such that for any pairwise distinct and nonzero elements $a,b,c\in R$, $ab+bc+ca$ is invertible. [i]Vlad Matei[/i]

1999 Brazil Team Selection Test, Problem 4
Tags: function , logarithms , inequalities , induction , number theory , Ring Theory , triangle inequality
Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

2006 IMC, 3
Tags: linear algebra , matrix , Ring Theory , IMC , college contests
Let $A$ be an $n$x$n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n$x$n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $detB_{i}=b_{i}$ for all $i=1,...,k$.

2003 Romania National Olympiad, 1
Tags: linear algebra , abstract algebra , Ring Theory , Isomorphisms
[b]a)[/b] Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field. [b]b)[/b] Prove that $$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$ for any natural number $ n\ge 2. $ [i]Marian Andronache, Ion Sava[/i]

2000 IMC, 5
Tags: superior algebra , superior algebra solved , Ring Theory
Let $R$ be a ring of characteristic zero. Let $e,f,g\in R$ be idempotent elements (an element $x$ is called idempotent if $x^2=x$) satisfying $e+f+g=0$. Show that $e=f=g=0$.

1993 All-Russian Olympiad, 4
Tags: induction , function , algebra , polynomial , Ring Theory , vector , binomial theorem
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

2011 Romania National Olympiad, 3
Tags: abstract algebra , Ring Theory
The equation $ x^{n+1} +x=0 $ admits $ 0 $ and $ 1 $ as its unique solutions in a ring of order $ n\ge 2. $ Prove that this ring is a skew field.

1987 Traian Lălescu, 2.1
Tags: abstract algebra , Ring Theory , extension rings , superrings , algebra , polynomial
Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.

Russian TST 2017, P3
Tags: algebra , polynomial , Ring Theory , Field theory , Miklos Schweitzer
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

1965 Miklós Schweitzer, 2
Tags: vector , algebra , polynomial , induction , Ring Theory , superior algebra , superior algebra unsolved
Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).

1973 Miklós Schweitzer, 2
Tags: Ring Theory , superior algebra , superior algebra unsolved
Let $ R$ be an Artinian ring with unity. Suppose that every idempotent element of $ R$ commutes with every element of $ R$ whose square is $ 0$. Suppose $ R$ is the sum of the ideals $ A$ and $ B$. Prove that $ AB\equal{}BA$. [i]A. Kertesz[/i]

2022 Romania National Olympiad, P2
Tags: Ring Theory , romania
Determine all rings $(A,+,\cdot)$ such that $x^3\in\{0,1\}$ for any $x\in A.$ [i]Mihai Opincariu[/i]

1998 Belarus Team Selection Test, 2
Tags: inequalities , Ring Theory , inequalities proposed , 109 , Belarus , algebra
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

2006 VJIMC, Problem 1
Tags: Ring Theory , abstract algebra
(a) Let $u$ and $v$ be two nilpotent elements in a commutative ring (with or without unity). Prove that $u+v$ is also nilpotent. (b) Show an example of a (non-commutative) ring $R$ and nilpotent elements $u,v\in R$ such that $u+v$ is not nilpotent.

2016 Miklós Schweitzer, 3
Tags: algebra , polynomial , Ring Theory , Field theory , Miklos Schweitzer
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.