Olympiad problems

2014 Miklós Schweitzer, 5

Tags: superior algebra

Let $ \alpha $ be a non-real algebraic integer of degree two, and let $ \mathbb{P} $ be the set of irreducible elements of the ring $ \mathbb{Z}[ \alpha] $. Prove that \[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]

2008 IberoAmerican Olympiad For University Students, 7

Tags: inequalities , superior algebra , abstract algebra , group theory

Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$). Prove that each subgroup of $A$ of finite index is isomorphic to $A$.

2006 District Olympiad, 3

Tags: algebra , polynomial , superior algebra

Prove that if $A$ is a commutative finite ring with at least two elements and $n$ is a positive integer, then there exists a polynomial of degree $n$ with coefficients in $A$ which does not have any roots in $A$.

2002 District Olympiad, 1

Tags: superior algebra , group theory

Let $ A $ be a ring, $ a\in A, $ and let $ n,k\ge 2 $ be two natural numbers such that $ n\vdots\text{char} (A) $ and $ 1+a=a^k. $ Show that the following propositions are true: [b]a)[/b] $ \forall s\in\mathbb{N}\quad \exists p_0,p_1,\ldots ,p_{k-1}\in\mathbb{Z}_{\ge 0}\quad a^s=\sum_{i=0}^{k-1} p_ia^{i} . $ [b]b)[/b] $ \text{ord} (a)\neq\infty . $

2005 Romania National Olympiad, 4

Tags: topology , induction , ring theory , superior algebra

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]

2003 District Olympiad, 3

Tags: quadratic , superior algebra , polynomial , algebra

Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that: (a) $1+1 \neq 0$; (b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$. [i]Marian Andronache[/i] [Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful. [Edit $2^\circ$] OK, thanks.

2021 Science ON all problems, 2

Tags: superior algebra , abstract algebra , rings

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

2021 Science ON all problems, 4

Tags: superior algebra , group theory

Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\ $\textbf{(a)}$ Prove that $G$ is cyclic. \\ $\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?

1962 Miklós Schweitzer, 2

Tags: superior algebra

Determine the roots of unity in the field of $ p$-adic numbers. [i]L. Fuchs[/i]

2014 IMS, 5

Tags: group theory , abstract algebra , superior algebra

Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomorphisms from $G_2$ to $H$. Prove that $G_1$ and $G_2$ are Isomorphic.

2005 District Olympiad, 3

Tags: superior algebra

Let $(G,\cdot)$ be a group and let $F$ be the set of elements in the group $G$ of finite order. Prove that if $F$ is finite, then there exists a positive integer $n$ such that for all $x\in G$ and for all $y\in F$, we have \[ x^n y = yx^n. \]

2019 Romania National Olympiad, 4

Tags: polynomial , complex number , root , superior algebra

Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that $\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct. $\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$

1996 Romania National Olympiad, 1

Tags: superior algebra , group theory , group isomorphism , group

Prove that a group $G$ in which exactly two elements other than the identity commute with each other is isomorphic to $\mathbb{Z}/3 \mathbb{Z}$ or $S_3.$

2009 District Olympiad, 4

Tags: probability , superior algebra , polynomial , finite field

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

2002 Romania National Olympiad, 4

Tags: algebra , polynomial , superior algebra

Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that: $a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$; $b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.

1998 IMC, 2

Tags: rotation , superior algebra , geometric transformation , geometry

Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$. (a) Prove the statement for $n=3$ and $n=5$. (b) Disprove the statement for $n=4$.

2001 District Olympiad, 1

Tags: abstract algebra , superior algebra , group theory , pigeonhole principle

For any $n\in \mathbb{N}^*$, let $H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}$. a) Prove that $H_n$ is a subgroup of the group $(Q,+)$ and that $Q=\bigcup_{n\in \mathbb{N}^*} H_n$; b) Prove that if $G_1,G_2,\ldots, G_m$ are subgroups of the group $(Q,+)$ and $G_i\neq Q,\ (\forall) 1\le i\le m$, then $G_1\cup G_2\cup \ldots \cup G_m\neq Q$ [i]Marian Andronache & Ion Savu[/i]

1985 Traian Lălescu, 1.3

Tags: superior algebra , group theory , abstract algebra

Let $ G $ be a finite group of odd order having, at least, three elements. For $ a\in G $ denote $ n(a) $ as the number of ways $ a $ can be written as a product of two distinct elements of $ G. $ Prove that $ \sum_{\substack{a\in G\\a\neq\text{id}}} n(a) $ is a perfect square.

1998 Romania National Olympiad, 4

Tags: superior algebra

Let $K\subseteq \mathbb C$ be a field with the operations from $\mathbb C$ s.t. i) K has exactly two endomorphisms, namely f and g ii) if f(x)=g(x) then $x\in\mathbb Q$. Prove that there exists an integer $d\neq 1$ free from squares so that $K=\mathbb Q(\sqrt d)$.

2013 Miklós Schweitzer, 3

Tags: abstract algebra , group theory , superior algebra

Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$-Sylow subgroup of $A_n$. [i]Proposed by Péter Pál Pálfy[/i]

1970 Miklós Schweitzer, 1

Tags: superior algebra

We have $ 2n\plus{}1$ elements in the commutative ring $ R$: \[ \alpha,\alpha_1,...,\alpha_n,\varrho_1,...,\varrho_n .\] Let us define the elements \[ \sigma_k\equal{}k\alpha \plus{} \sum_{i\equal{}1}^n \alpha_i\varrho_i^k .\] Prove that the ideal $ (\sigma_0,\sigma_1,...,\sigma_k,...)$ can be finitely generated. [i]L. Redei[/i]

1993 Hungary-Israel Binational, 7

Tags: abstract algebra , group theory , superior algebra

In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Assume $|G'| = 2$. Prove that $|G : G'|$ is even.

1997 Romania National Olympiad, 3

Tags: polynomial , superior algebra , field theory , algebra

Let $K$ be a finite field, $n \ge 2$ an integer, $f \in K[X]$ an irreducible polynomial of degree $n,$ and $g$ the product of all the nonconstant polynomials in $K[X]$ of degree at most $n-1.$ Prove that $f$ divides $g-1.$

2011 District Olympiad, 4

Tags: abstract algebra , ring theory , nilpotence , cardinal , superior algebra , group theory

Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove: [b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $ [b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $

2010 District Olympiad, 2

Tags: group theory , superior algebra , abstract algebra

Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$. i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian. ii) Give an example of a non-abelian group with $ G$'s property from the enounce.