Olympiad problems

2000 District Olympiad (Hunedoara), 1

Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $ [b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication. [b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups? [b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $

1993 Hungary-Israel Binational, 5

Tags: group theory , abstract algebra , 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. Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?

2017 District Olympiad, 2

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

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.

2018 Brazil Undergrad MO, 24

Tags: summation , superior algebra

What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$

2006 Pre-Preparation Course Examination, 4

Tags: superior algebra , algebra , polynomial

Show that for every prime $p$ and integer $n$, there is an irreducible polynomial of degree $n$ in $\mathbb{Z}_p[x]$ and use that to show there is a field of size $p^n$.

2019 District Olympiad, 1

Tags: endomorphism , superior algebra , group , abstract algebra

Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$ $\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism. $\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?

1974 Miklós Schweitzer, 4

Tags: superior algebra , ring theory , function , induction , number theory , prime number

Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

1986 Traian Lălescu, 2.4

Tags: group theory , abstract algebra , superior algebra

Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $

1987 Traian Lălescu, 1.1

Tags: group theory , superior algebra

Describe all groups $ G $ which have the property that: $$ (\forall H\le G)(\forall x,y\in G)(xy\in H\implies (x,y\in H\vee xy=1)) $$

2009 IMS, 7

Tags: abstract algebra , group theory , superior algebra

Let $ G$ be a group such that $ G'$ is abelian and each normal and abelian subgroup of $ G$ is finite. Prove that $ G$ is finite.

2007 District Olympiad, 4

Tags: superior algebra , algebra , polynomial , quadratic

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]

1970 Miklós Schweitzer, 2

Tags: superior algebra , abstract algebra

Let $ G$ and $ H$ be countable Abelian $ p$-groups ($ p$ an arbitrary prime). Suppose that for every positive integer $ n$, \[ p^nG \not\equal{} p^{n\plus{}1}G .\] Prove that $ H$ is a homomorphic image of $ G$. [i]M. Makkai[/i]

2013 Romania National Olympiad, 2

Tags: superior algebra , polynomial , algebra

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

1962 Miklós Schweitzer, 2

Tags: superior algebra

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

2010 IMC, 3

Tags: group theory , abstract algebra , induction , strong induction , superior algebra

Denote by $S_n$ the group of permutations of the sequence $(1,2,\dots,n).$ Suppose that $G$ is a subgroup of $S_n,$ such that for every $\pi\in G\setminus\{e\}$ there exists a unique $k\in \{1,2,\dots,n\}$ for which $\pi(k)=k.$ (Here $e$ is the unit element of the group $S_n.$) Show that this $k$ is the same for all $\pi \in G\setminus \{e\}.$

2019 Romania National Olympiad, 2

Tags: group theory , abstract algebra , superior algebra

Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$

1993 Hungary-Israel Binational, 4

Tags: superior algebra , group theory , abstract 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. Let $H \leq G$ and $a, b \in G.$ Prove that $|aH \cap Hb|$ is either zero or a divisor of $|H |.$

2007 IMS, 1

Tags: superior algebra , group theory , abstract algebra

Suppose there exists a group with exactly $n$ subgroups of index 2. Prove that there exists a finite abelian group $G$ that has exactly $n$ subgroups of index 2.

1999 Romania National Olympiad, 2

Tags: group theory , abstract algebra , superior algebra

For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it. Decide whether the following statements are true or false. a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$ b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$

2005 Romania National Olympiad, 4

Tags: superior algebra , induction , topology , ring theory

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]

2006 District Olympiad, 3

Tags: superior algebra , polynomial , 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 $A$.

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 Romania National Olympiad, 4

Tags: polynomial , superior algebra , algebra

[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients. [/color]

1969 Miklós Schweitzer, 2

Tags: superior algebra

Let $ p\geq 7$ be a prime number, $ \zeta$ a primitive $ p$th root of unity, $ c$ a rational number. Prove that in the additive group generated by the numbers $ 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3}$ there are only finitely many elements whose norm is equal to $ c$. (The norm is in the $ p$th cyclotomic field.) [i]K. Gyory[/i]

2018 Romania National Olympiad, 4

Tags: polynomial , irreducible , superior algebra

For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$ [i]Marius Vladoiu[/i]