Olympiad problems

2008 IMS, 5

Prove that there does not exist a ring with exactly 5 regular elements. ($ a$ is called a regular element if $ ax \equal{} 0$ or $ xa \equal{} 0$ implies $ x \equal{} 0$.) A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.

2004 Alexandru Myller, 1

Tags: abstract algebra , Field theory , Ring Theory , freshman s dream , number theory

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]

2003 Romania National Olympiad, 4

Tags: abstract algebra , group theory , Ring Theory , limits

$ 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]

2015 Romania National Olympiad, 1

Tags: superior algebra , Ring Theory , idempotency

Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that: [b]a)[/b] $ 1 $ is the only unit of this ring. [b]b)[/b] this ring is Boolean.

2012 Iran MO (3rd Round), 4

Tags: algebra , polynomial , Ring Theory , modular arithmetic , number theory proposed , number theory

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

2014 Contests, 3

Tags: vector , abstract algebra , Ring Theory , superior algebra , superior algebra unsolved

Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.

2012 Korea - Final Round, 3

Tags: analytic geometry , algebra , polynomial , Ring Theory , combinatorics unsolved , combinatorics

Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

1951 Miklós Schweitzer, 10

Tags: algebra , polynomial , modular arithmetic , Ring Theory

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

2010 Romania National Olympiad, 2

Tags: algebra , polynomial , Ring Theory , superior algebra , superior algebra unsolved

We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element. a) If in $A$, $1+1=0$, prove that $A$ has property $(P)$ if and only if $A$ is a field. b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$. [i]Dan Schwarz[/i]

2001 VJIMC, Problem 4

Tags: Ring Theory , abstract algebra

Let $R$ be an associative non-commutative ring and let $n>2$ be a fixed natural number. Assume that $x^n=x$ for all $x\in R$. Prove that $xy^{n-1}=y^{n-1}x$ holds for all $x,y\in R$.

2005 VJIMC, Problem 4

Tags: Ring Theory , abstract algebra

Let $R$ ba a finite ring with the following property: for any $a,b\in R$ there exists an element $c\in R$ (depending on $a$ and $b$) such that $a^2+b^2=c^2$. Prove that for any $a,b,c\in R$ there exists $d\in R$ such that $2abc=d^2$. (Here $2abc$ denotes $abc+abc$. The ring $R$ is assumed to be associative, but not necessarily commutative and not necessarily containing a unit.

2009 Romania National Olympiad, 2

Tags: Ring Theory , abstract algebra , Nilpotent , algebra , Commutative Algebra

[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. [b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.

2002 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratics , Ring Theory , equation , functional equation

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

1990 Vietnam Team Selection Test, 1

Tags: Ring Theory , number theory , relatively prime , greatest common divisor , least common multiple , number theory unsolved

Let $ T$ be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$. 2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$. For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.

2009 Romania National Olympiad, 3

Tags: abstract algebra , Ring Theory , number theory

Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares.

2015 Romania National Olympiad, 2

Tags: Ring Theory , Field theory , superior algebra

Show that the set of all elements minus $ 0 $ of a finite division ring that has at least $ 4 $ elements can be partitioned into two nonempty sets $ A,B $ having the property that $$ \sum_{x\in A} x=\prod_{y\in B} y. $$

2012 VJIMC, Problem 3

Tags: Ring Theory , abstract algebra , group theory

Let $(A,+,\cdot)$ be a ring with unity, having the following property: for all $x\in A$ either $x^2=1$ or $x^n=0$ for some $n\in\mathbb N$. Show that $A$ is a commutative ring.

1993 China Team Selection Test, 3

Tags: geometry , circumcircle , incenter , trigonometry , geometric transformation , reflection , Ring Theory

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2016 District Olympiad, 1

Tags: superior algebra , Ring Theory , abstract algebra , group theory , FTA

A ring $ A $ has property [i](P),[/i] if $ A $ is finite and there exists $ (\{ 0\}\neq R,+)\le (A,+) $ such that $ (U(A),\cdot )\cong (R,+) . $ Show that: [b]a)[/b] If a ring has property [i](P),[/i] then, the number of its elements is even. [b]b)[/b] There are infinitely many rings of distinct order that have property [i](P).[/i]

2010 Laurențiu Panaitopol, Tulcea, 4

Tags: abstract algebra , Ring Theory , polynomial , polynomial ring , beautiful

Let be a ring $ R $ which has the property that there exist two distinct natural numbers $ s,t $ such that for any element $ x $ of $ R, $ the equation $ x^s=x^t $ is true. Show that there exists a polynom in $ R[X] $ of degree $$ |s-t|\left( 1+|s-t| \right) $$ such that all the elements of $ R $ are roots of it.

2015 District Olympiad, 4

Tags: Ring Theory , abstract algebra , nilpotence , superior algebra

Let $ m $ be a non-negative ineger, $ n\ge 2 $ be a natural number, $ A $ be a ring which has exactly $ n $ elements, and an element $ a $ of $ A $ such that $ 1-a^k $ is invertible, for all $ k\in\{ m+1,m+2,...,m+n-1\} . $ Prove that $ a $ is nilpotent.

2002 VJIMC, Problem 2

Tags: Ring Theory , abstract algebra

A ring $R$ (not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that $R$ is finite.

2008 Alexandru Myller, 4

Tags: Ring Theory , Cauchy , injectivity , superior algebra

In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of $ 2. $ [i]Dinu Şerbănescu[/i]

2007 Grigore Moisil Intercounty, 3

Tags: Ring Theory

Let be a nontrivial finite ring having the property that any element of it has an even power that is equal to itself. Prove that [b]a)[/b] the order of the ring is a power of $ 2. $ [b]b)[/b] the sum of all elements of the ring is $ 0. $

2014 PUMaC Individual Finals A, 1

Tags: geometry , incenter , circumcircle , Ring Theory , geometric transformation , analytic geometry , barycentric

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.