Found problems: 84
1965 Miklós Schweitzer, 2
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$).
2002 VJIMC, Problem 2
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.
1999 Brazil Team Selection Test, Problem 4
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+.
1998 Belarus Team Selection Test, 2
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]
2012 Iran MO (3rd Round), 4
$P(x)$ 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]
2003 SNSB Admission, 3
Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.
2015 Romania National Olympiad, 2
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
$$ \sum_{x\in A} x=\prod_{y\in B} y. $$
2003 Romania National Olympiad, 1
[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]
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.
2012 Korea - Final Round, 3
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 $y\mid x$.
2007 IMS, 6
Let $R$ be a commutative ring with 1. Prove that $R[x]$ has infinitely many maximal ideals.
2011 Romania National Olympiad, 3
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.
2014 IMS, 3
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.
1993 China Team Selection Test, 3
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.
2010 Laurențiu Panaitopol, Tulcea, 4
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.
1966 Miklós Schweitzer, 4
Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that
a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and
b) $ I$ contains of a polynomial with constant term $ 1$.
Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$.
[i]Gy. Szekeres[/i]
2014 Contests, 3
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 VJIMC, Problem 3
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.
2006 MOP Homework, 7
Let $A_{n,k}$ denote the set of lattice paths in the coordinate plane of upsteps $u=[1,1]$, downsteps $d=[1,-1]$, and flatsteps $f=[1,0]$ that contain $n$ steps, $k$ of which are slanted ($u$ or $d$). A sharp turn is a consecutive pair of $ud$ or $du$. Let $B_{n,k}$ denote the set of paths in $A_{n,k}$ with no upsteps among the first $k-1$ steps, and let $C_{n,k}$ denote the set of paths in $A_{n,k}$ with no sharps anywhere. For example, $fdu$ is in $B_{3,2}$ but not in $C_{3,2}$, while $ufd$ is in $C_{3,2}$ but not $B_{3,2}$. For $1 \le k \le n$, prove that the sets $B_{n,k}$ and $C_{n,k}$ contains the same number of elements.
1976 Miklós Schweitzer, 4
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]
2015 District Olympiad, 4
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.
2001 Miklós Schweitzer, 4
Find the units of $R=\mathbb Z[t][\sqrt{t^2-1}]$.
2006 IMC, 3
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 $i=1,...,k$.
2005 Romania National Olympiad, 4
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]
2022 Romania National Olympiad, P2
Determine all rings $(A,+,\cdot)$ such that $x^3\in\{0,1\}$ for any $x\in A.$
[i]Mihai Opincariu[/i]