Olympiad problems

1989 Greece National Olympiad, 4

In a group $G$, we have two elements $x,y$ such that $x^{n}=e,y^2=e,yxy=x^{-1}$, $n\ge 1$. Prove that for any $k\in\mathbb{N}$ holds $(x^ky)^2=e$. Note : e=group's identity .

2012 District Olympiad, 3

Tags: function , greatest common divisor , superior algebra

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

2000 IMC, 5

Tags: ring theory , superior algebra

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 $e=f=g=0$.

1996 Romania National Olympiad, 3

Tags: superior algebra , ring theory , endomorphism

Let $A$ be a commutative ring with $0 \neq 1$ such that for any $x \in A \setminus \{0\}$ there exist positive integers $m,n$ such that $(x^m+1)^n=x.$ Prove that any endomorphism of $A$ is an automorphism.

1975 Miklós Schweitzer, 3

Tags: superior algebra

Let $ S$ be a semigroup without proper two-sided ideals and suppose that for every $ a,b \in S$ at least one of the products $ ab$ and $ ba$ is equal to one of the elements $ a,b$. Prove that either $ ab\equal{}a$ for all $ a,b \in S$ or $ ab\equal{}b$ for all $ a,b \in S$. [i]L. Megyesi[/i]

1999 Romania National Olympiad, 4

Tags: superior algebra , integral domain , polynomial ring

Let $A$ be an integral domain and $A[X]$ be its associated ring of polynomials. For every integer $n \ge 2$ we define the map $\varphi_n : A[X] \to A[X],$ $\varphi_n(f)=f^n$ and we assume that the set $$M= \Big\{ n \in \mathbb{Z}_{\ge 2} : \varphi_n \mathrm{~is~an~endomorphism~of~the~ring~} A[X] \Big\}$$ is nonempty. Prove that there exists a unique prime number $p$ such that $M=\{p,p^2,p^3, \ldots\}.$

1966 Miklós Schweitzer, 8

Tags: polynomial , function , ring theory , superior algebra , algebra

Prove that in Euclidean ring $ R$ the quotient and remainder are always uniquely determined if and only if $ R$ is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases: $ R$ can also be a field or the null ring.) [i]E. Fried[/i]

1970 IMO Longlists, 28

Tags: function , superior algebra

A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled: $(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$. $(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$. $(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x = v$, and an element $y\in G$ such that $y\circ u = v$. Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.