Found problems: 250
2002 Iran Team Selection Test, 12
We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic.
[i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.
2005 VJIMC, Problem 2
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
1996 Turkey MO (2nd round), 2
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
2011 Spain Mathematical Olympiad, 2
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
2019 District Olympiad, 3
Let $G$ be a finite group and let $x_1,…,x_n$ be an enumeration of its elements. We consider the matrix $(a_{ij})_{1 \le i,j \le n},$ where $a_{ij}=0$ if $x_ix_j^{-1}=x_jx_i^{-1},$ and $a_{ij}=1$ otherwise. Find the parity of the integer $\det(a_{ij}).$
2008 Vietnam Team Selection Test, 3
Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:
$ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$;
$ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$.
Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).
2006 Romania National Olympiad, 3
Let $\displaystyle G$ be a finite group of $\displaystyle n$ elements $\displaystyle ( n \geq 2 )$ and $\displaystyle p$ be the smallest prime factor of $\displaystyle n$. If $\displaystyle G$ has only a subgroup $\displaystyle H$ with $\displaystyle p$ elements, then prove that $\displaystyle H$ is in the center of $\displaystyle G$.
[i]Note.[/i] The center of $\displaystyle G$ is the set $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.
2008 IMS, 5
Prove that there does not exist a ring with exactly 5 regular elements.
($ a$ is called a regular element if $ ax \equal{} 0$ or $ xa \equal{} 0$ implies $ x \equal{} 0$.)
A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.
1994 Miklós Schweitzer, 2
For which finite group G does there exist natural number s with the following property: for any subgroup H of a finite direct power of G, each subgroup of H is produced as an intersection of subgroups of H with index at most s.
not sure of translation.
2008 IMC, 5
Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\text{Aut } H| > |\text{Aut } G|$? Disprove or provide an example. Here the notation $ |\text{Aut } X|$ for some group $ X$ denotes the number of isomorphisms from $ X$ to itself.
2007 Iran MO (3rd Round), 5
A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property:
For each $ a\in\mathbb N$, that $ (a,m) \equal{} 1$, has a unique representation in the following form:
\[ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}\]
Prove that for each $ m$ we have a hyper-primitive root.
2007 Purple Comet Problems, 15
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.
2013 Iran Team Selection Test, 2
Find the maximum number of subsets from $\left \{ 1,...,n \right \}$ such that for any two of them like $A,B$ if $A\subset B$ then $\left | B-A \right |\geq 3$. (Here $\left | X \right |$ is the number of elements of the set $X$.)
1969 Miklós Schweitzer, 3
Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\]
[i]A. Mate[/i]
2004 Nicolae Coculescu, 1
Let $ b,a $ be two elements of a group chosen such that $ a^3b=ba^2 $ and $ a^2b=ba^3. $ Prove that the order of $ a $ divides $ 5. $
[i]Gabriel Tica[/i]
1977 IMO Shortlist, 13
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
2009 Miklós Schweitzer, 5
Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then
(i) $ G$ is not simple;
(ii) $ G$ contains a normal subgroup of order $ m$.
2023 Romania National Olympiad, 1
Let $(G, \cdot)$ a finite group with order $n \in \mathbb{N}^{*},$ where $n \geq 2.$ We will say that group $(G, \cdot)$ is arrangeable if there is an ordering of its elements, such that
\[
G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}.
\]
a) Determine all positive integers $n$ for which the group $(Z_n, +)$ is arrangeable.
b) Give an example of a group of even order that is arrangeable.
2010 IMC, 3
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 $\pi \in G\setminus \{e\}.$
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]
2008 Gheorghe Vranceanu, 2
Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.
2012 VJIMC, Problem 3
Let $(A,+,\cdot)$ be a ring with unity, having the following property: for all $x\in A$ either $x^2=1$ or $x^n=0$ for some $n\in\mathbb N$. Show that $A$ is a commutative ring.
1985 Miklós Schweitzer, 6
Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]
2021 IMC, 6
For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.
2009 Putnam, A5
Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$