Found problems: 339
Gheorghe Țițeica 2024, P2
a) Let $n$ be a positive integer $G$ be a a group with $|G|<\frac{4n^2}{n-\varphi(n)}$. Suppose that $Z(G)$ contains at least $\varphi(n)+1$ elements of order $n$. Prove that $G$ is abelian.
b) Find a noncommutative group $G$ with $16$ elements such that $Z(G)$ contains two elements of order two.
[i]Robert Rogozsan & Filip Munteanu[/i]
2022 Bulgarian Spring Math Competition, Problem 12.4
Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$.
2021 Alibaba Global Math Competition, 16
Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$, one has that
\[\text{dim} V^{H_1}=\text{dim} V^{H_2},\]
where $V^{H_i}$ is the subspace of $H_i$-invariant vectors in $V$ ($i=1,2$). Prove that
\[Z(G) \cap H_1=Z(G) \cap H_2.\]
Here $Z(G)$ denotes the center of $G$.
1981 Miklós Schweitzer, 7
Let $ U$ be a real normed space such that, for an finite-dimensional, real normed space $ X,U$ contains a subspace isometrically isomorphic to $ X$. Prove that every (not necessarily closed) subspace $ V$ of $ U$ of finite codimension has the same property. (We call $ V$ of finite codimension if there exists a finite-dimensional subspace $ N$ of $ U$ such that $ V\plus{}N\equal{}U$.)
[i]A. Bosznay[/i]
2009 Romania National Olympiad, 2
[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring.
[b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.
2010 Romania National Olympiad, 3
Let $G$ be a finite group of order $n$. Define the set
\[H=\{x:x\in G\text{ and }x^2=e\},\]
where $e$ is the neutral element of $G$. Let $p=|H|$ be the cardinality of $H$. Prove that
a) $|H\cap xH|\ge 2p-n$, for any $x\in G$, where $xH=\{xh:h\in H\}$.
b) If $p>\frac{3n}{4}$, then $G$ is commutative.
c) If $\frac{n}{2}<p\le\frac{3n}{4}$, then $G$ is non-commutative.
[i]Marian Andronache[/i]
2010 Iran MO (3rd Round), 3
suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)
1969 Canada National Olympiad, 7
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
2007 Nicolae Coculescu, 3
Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition:
$$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$
[i]Marius Perianu[/i]
2012 Tuymaada Olympiad, 4
Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the [i]weight of a vertex [/i]the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero.
[i]Proposed by W. Tutte[/i]
2010 Iran MO (3rd Round), 2
prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)
2006 Germany Team Selection Test, 3
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
2003 Alexandru Myller, 3
Let be three elements $ a,b,c $ of a nontrivial, noncommutative ring, that satisfy $ ab=1-c, $ and such that there exists an element $ d $ from the ring such that $ a+cd $ is a unit. Prove that there exists an element $ e $ from the ring such that $ b+ec $ is a unit.
[i]Andrei Nedelcu[/i] and [i] Lucian Ladunca [/i]
2010 N.N. Mihăileanu Individual, 4
If $ p $ is an odd prime, then the following characterization holds.
$$ 2^{p-1}\equiv 1\pmod{p^2}\iff \sum_{2=q}^{(p-1)/2} q^{p-2}\equiv -1\pmod p $$
[i]Marius Cavachi[/i]
2024 Assara - South Russian Girl's MO, 3
In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $2$?
[i]G.M.Sharafetdinova[/i]
2006 Cezar Ivănescu, 2
Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.
2013 China National Olympiad, 3
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.
1957 Miklós Schweitzer, 10
[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]
1978 Miklós Schweitzer, 2
For a distributive lattice $ L$, consider the following two statements:
(A) Every ideal of $ L$ is the kernel of at least two different homomorphisms.
(B) $ L$ contains no maximal ideal.
Which one of these statements implies the other?
(Every homomorphism $ \varphi$ of $ L$ induces an equivalence relation on $ L$: $ a \sim b$ if and only if $ a \varphi\equal{}b \varphi$. We do not consider two homomorphisms different if they imply the same equivalence relation.)
[i]J. Varlet, E. Fried[/i]
PEN M Problems, 27
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.
1974 Spain Mathematical Olympiad, 5
Let $(G, \cdot )$ be a group and $e$ an identity element. Prove that if all elements $x$ of $G$ satisfy $x\cdot x = e$ then $(G, \cdot)$ is abelian (that is, commutative).
1966 Miklós Schweitzer, 4
Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that
a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and
b) $ I$ contains of a polynomial with constant term $ 1$.
Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$.
[i]Gy. Szekeres[/i]
2001 VJIMC, Problem 4
Let $R$ be an associative non-commutative ring and let $n>2$ be a fixed natural number. Assume that $x^n=x$ for all $x\in R$. Prove that $xy^{n-1}=y^{n-1}x$ holds for all $x,y\in R$.
2007 Bulgaria National Olympiad, 3
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
1984 Putnam, B3
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$,
$(\text i)$ $x*z=y*z$ implies $x=y$
$(\text{ii})$ $x*(y*z)\ne(x*y)*z$