Found problems: 339
2002 Iran Team Selection Test, 10
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
2010 IMC, 4
Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.
2008 Argentina National Olympiad, 2
In every cell of a $ 60 \times 60$ board is written a real number, whose absolute value is less or equal than $ 1$. The sum of all numbers on the board equals $ 600$.
Prove that there is a $ 12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $ 24$.
2001 Miklós Schweitzer, 3
How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?
2009 IberoAmerican Olympiad For University Students, 7
Let $G$ be a group such that every subgroup of $G$ is subnormal. Suppose that there exists $N$ normal subgroup of $G$ such that $Z(N)$ is nontrivial and $G/N$ is cyclic. Prove that $Z(G)$ is nontrivial. ($Z(G)$ denotes the center of $G$).
[b]Note[/b]: A subgroup $H$ of $G$ is subnormal if there exist subgroups $H_1,H_2,\ldots,H_m=G$ of $G$ such that $H\lhd H_1\lhd H_2 \lhd \ldots \lhd H_m= G$ ($\lhd$ denotes normal subgroup).
2014 VJIMC, Problem 2
Let $p$ be a prime number and let $A$ be a subgroup of the multiplicative group $\mathbb F^*_p$ of the finite field $\mathbb F_p$ with $p$ elements. Prove that if the order of $A$ is a multiple of $6$, then there exist $x,y,z\in A$ satisfying $x+y=z$.
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 $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.
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.
2011 Putnam, A6
Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$
Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.
2009 Putnam, B1
Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$
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 $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.
2010 Iran MO (3rd Round), 5
suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$.
SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.
the exam time was 5 hours.
1993 Hungary-Israel Binational, 7
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.
Assume $|G'| = 2$. Prove that $|G : G'|$ is even.
2025 Korea Winter Program Practice Test, P7
There are $2025$ positive integers $a_1, a_2, \cdots, a_{2025}$ are placed around a circle. For any $k = 1, 2, \cdots, 2025$, $a_k \mid a_{k-1} + a_{k+1}$ where indices are considered modulo $n$. Prove that there exists a positive integer $N$ such that satisfies the following condition.
[list]
[*] [b](Condition)[/b] For any positive integer $n > N$, when $a_1 = n^n$, $a_1, a_2, \cdots, a_{2025}$ are all multiples of $n$.
[/list]
2013 Romania National Olympiad, 4
Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.
2007 Nicolae Coculescu, 1
Let be the set $ G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} $ and a function $ \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} $ having the property that the operation $ *:G^2\longrightarrow G $ defined as
$$ (a,b)*(c,d)=(ac,bc+d\varphi (a)) $$
is associative.
[b]a)[/b] Show that $ (G,*) $ is a group.
[b]b)[/b] Describe $ \varphi , $ knowing that $(G,*) $ is a commutative group.
[i]Marius Perianu[/i]
2016 Romania National Olympiad, 4
Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$
[i]Marian Andronache[/i]
2012 Bulgaria National Olympiad, 1
Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.
2013 Albania Team Selection Test, 5
Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that :
$(2^k)!=2^n*m$
2014 Baltic Way, 18
Let $p$ be a prime number, and let $n$ be a positive integer. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i\in \{0, 1, \ldots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that \[p^n \mid (a_1a_2 + a_3a_4 + 1).\]
Gheorghe Țițeica 2024, P3
Determine all commutative rings $R$ with at least four elements that are not fields, such that for any pairwise distinct and nonzero elements $a,b,c\in R$, $ab+bc+ca$ is invertible.
[i]Vlad Matei[/i]
2004 VJIMC, Problem 1
Are the groups $(\mathbb Q,+)$ and $(\mathbb Q^+,\cdot)$ isomorphic?
2005 IMO Shortlist, 7
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]
2014 South africa National Olympiad, 1
Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.
2009 District Olympiad, 2
Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $