Found problems: 183
2001 District Olympiad, 2
Let $K$ commutative field with $8$ elements. Prove that $(\exists)a\in K$ such that $a^3=a+1$.
[i]Mircea Becheanu[/i]
2014 IMC, 2
Let $A=(a_{ij})_{i, j=1}^n$ be a symmetric $n\times n$ matrix with real entries, and let $\lambda _1, \lambda _2, \dots, \lambda _n$ denote its eigenvalues. Show that
$$\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j$$
and determine all matrices for which equality holds.
(Proposed by Matrin Niepel, Comenius University, Bratislava)
2013 Miklós Schweitzer, 4
Let $A$ be an Abelian group with $n$ elements. Prove that there are two subgroups in $\text{GL}(n,\Bbb{C})$, isomorphic to $S_n$, whose intersection is isomorphic to the automorphism group of $A$.
[i]Proposed by Zoltán Halasi[/i]
2013 District Olympiad, 2
Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any
automorphism f for G,there are two automorphisms
g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that:
(a) Every group which the property $\left( P \right)$ is comutative.
(b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property.
(c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property.
(The order of a finite group is the number of elements of that group).
2008 Alexandru Myller, 4
In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of $ 2. $
[i]Dinu Şerbănescu[/i]
1983 Miklós Schweitzer, 8
Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.)
[i]A. Huhn[/i]
1968 Miklós Schweitzer, 1
Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group $ G$. Prove that this ring is Neumann-regular if and only if $ G$ is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring $ R$ is called Neumann-regular if for every $ \alpha \in R$ there exists a $ \beta \in R$ such that $ \alpha \beta \alpha\equal{}\alpha$.)
[i]E. Freid[/i]
2007 IMS, 6
Let $R$ be a commutative ring with 1. Prove that $R[x]$ has infinitely many maximal ideals.
2005 Romania National Olympiad, 1
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]
2002 Romania National Olympiad, 4
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 $K[X]$.
2015 Miklos Schweitzer, 6
Let $G$ be the permutation group of a finite set $\Omega$.Consider $S\subset G$ such that $1\in S$ and for any $x,y\in \Omega$ there exists a unique element $\sigma \in S$ such that $\sigma (x)=y$.Prove that,if the elements of $S \setminus \{1\}$ are conjugate in $G$,then $G$ is $2-$transitive on $\Omega$
2001 Romania National Olympiad, 2
Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.
2014 Miklós Schweitzer, 5
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
\[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]
2021 Science ON all problems, 4
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?
2013 District Olympiad, 4
Problem 4. Let$\left( A,+,\cdot \right)$ be a ring with the property that $x=0$ is the only solution of the ${{x}^{2}}=0,x\in A$ecuation. Let $B=\left\{ a\in A|{{a}^{2}}=1 \right\}$. Prove that:
(a) $ab-ba=bab-a$, whatever would be $a\in A$ and $b\in B$.
(b) $\left( B,\cdot \right)$ is a group
2007 IberoAmerican Olympiad For University Students, 6
Let $F$ be a field whose characteristic is not $2$, let $F^*=F\setminus\left\{0\right\}$ be its multiplicative group and let $T$ be the subgroup of $F^*$ constituted by its finite order elements. Prove that if $T$ is finite, then $T$ is cyclic and its order is even.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3
Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.
1991 Putnam, B2
Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.
2012 Romania National Olympiad, 2
[color=darkred] Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that:
[list]
[b]a)[/b] $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$
[b]b)[/b] If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative.
[/list]
[/color]
2018 Romania National Olympiad, 1
Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$
2004 Romania National Olympiad, 2
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]
2021 Science ON all problems, 2
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]
2014 District Olympiad, 4
Let $(G,\cdot)$ be a group with no elements of order 4, and let
$f:G\rightarrow G$ be a group morphism such that $f(x)\in\{x,x^{-1}\}$, for
all $x\in G$. Prove that either $f(x)=x$ for all $x\in G$, or $f(x)=x^{-1}$
for all $x\in G$.
1963 Miklós Schweitzer, 3
Let $ R\equal{}R_1\oplus R_2$ be the direct sum of the rings $ R_1$ and $ R_2$, and let $ N_2$ be the annihilator ideal of $ R_2$ (in $ R_2$). Prove that $ R_1$ will be an ideal in every ring $ \widetilde{R}$ containing $ R$ as an ideal if and only if the only homomorphism from $ R_1$ to $ N_2$ is the zero homomorphism. [Gy. Hajos]
2006 Romania National Olympiad, 1
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.