Olympiad problems

2005 Alexandru Myller, 2

Tags: linear algebra , matrix , algebra , polynomial , superior algebra , superior algebra solved

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

2006 District Olympiad, 3

Tags: algebra , polynomial , superior algebra , superior algebra solved

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 $A$.

2020 Candian MO, 5#

Tags: algebra , polynomial , induction , superior algebra , superior algebra solved , first college math post

If A,B are invertible and the set {Ak - Bk | k is a natural number} is finite , then there exists a natural number m such that Am = Bm.

2004 Romania National Olympiad, 2

Tags: algebra , polynomial , function , superior algebra , superior algebra solved

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

2000 IMC, 5

Tags: superior algebra , superior algebra solved , Ring Theory

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 $e=f=g=0$.

2006 District Olympiad, 2

Tags: linear algebra , matrix , group theory , abstract algebra , invariant , superior algebra , superior algebra solved

Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.

2006 Romania National Olympiad, 1

Tags: abstract algebra , algebra , polynomial , group theory , superior algebra , superior algebra solved

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent: (a) $\displaystyle 1+1=0$; (b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.

1999 IMC, 1

Tags: superior algebra , superior algebra solved

Let $R$ be a ring where $\forall a\in R: a^2=0$. Prove that $abc+abc=0$ for all $a,b,c\in R$.

2004 Gheorghe Vranceanu, 1

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

Let $(G,\cdot)$ be a group, and let $H_1,H_2$ be proper subgroups s.t. $H_1\cap H_2=\{e\}$, where $e$ is the identity element of $G$. They also have the following properties: [b]i)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_1\setminus\{e\}\Rightarrow xy\in H_2$ [b]ii)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_2\setminus\{e\}\Rightarrow xy\in H_1$ Prove that: [b]a)[/b] $|H_1|=|H_2|$ [b]b)[/b] $|G|=|H_1|\cdot |H_2|$

2005 Alexandru Myller, 1

Tags: linear algebra , matrix , superior algebra , superior algebra solved

Let $A,B\in M_2(\mathbb Z)$ s.t. $AB=\begin{pmatrix}1&2005\\0&1\end{pmatrix}$. Prove that there is a matrix $C\in M_2(\mathbb Z)$ s.t. $BA=C^{2005}$. [i]Dinu Serbanescu[/i]

1998 Romania National Olympiad, 4

Tags: superior algebra , superior algebra solved

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 $K=\mathbb Q(\sqrt d)$.

2005 District Olympiad, 3

Tags: superior algebra , superior algebra solved

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 \[ x^n y = yx^n. \]

2005 Romania National Olympiad, 1

Tags: superior algebra , superior algebra solved

Prove that the group morphisms $f: (\mathbb{C},+)\to(\mathbb{C},+)$ for which there exists a positive $\lambda$ such that $|f(z)| \leq \lambda |z|$ for all $z\in\mathbb{C}$, have the form \[ f(z) = \alpha z + \beta \overline{z} \] for some complex $\alpha$, $\beta$. [i]Cristinel Mortici[/i]

2005 Alexandru Myller, 2

2006 District Olympiad, 3

2020 Candian MO, 5#

2004 Romania National Olympiad, 2

2000 IMC, 5

2006 District Olympiad, 2

2006 Romania National Olympiad, 1

1999 IMC, 1

2004 Gheorghe Vranceanu, 1

2005 Alexandru Myller, 1

1998 Romania National Olympiad, 4

2005 District Olympiad, 3

2005 Romania National Olympiad, 1

2005 Alexandru Myller, 4

2006 IMS, 3

2007 IMS, 6