Found problems: 339
2022 China Team Selection Test, 3
Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and
(1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer;
(2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$, there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
1993 Hungary-Israel Binational, 5
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.
Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?
2012 Centers of Excellency of Suceava, 1
Let be a natural number $ n\ge 2, $ a group $ G $ and two elements of it $ e_1,e_2 $ such that $ e_2e_1x=xe_2e_1, $ for any element $ x $ of $ G. $
Prove that $ \left( e_1xe_2 \right)^n =e_1x^ne_2, $ for any element $ x $ of $ G, $ if and only if $ e_2e_1=\left( e_2e_1\right)^n. $
[i]Ion Bursuc[/i]
1995 Brazil National Olympiad, 2
Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.
2023 Macedonian Team Selection Test, Problem 5
Let $Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x]$ be a polynomial with integer coefficients. For an odd prime number $p$ we define the polynomial $Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}.$
Assume that there exist infinitely primes $p$ such that
$$\frac{Q_{p}(x)-Q(x)}{p}$$
is an integer for all $x \in \mathbb{Z}$. Determine the largest possible value of $Q(2023)$ over all such polynomials $Q$.
[i]Authored by Nikola Velov[/i]
2024 TASIMO, 6
We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$
2006 District Olympiad, 2
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.
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]
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).
2005 All-Russian Olympiad, 1
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.
1992 Miklós Schweitzer, 8
Let $F$ be a set of filters on X so that if $ \sigma, \tau \in F$ , $\forall S \in\sigma$ , $\forall T\in\tau$ , we have $S \cap T\neq\emptyset$ , then $\sigma \cap \tau \in F$. We say that $F$ is compatible with a topology on X when $x \in X$ is a contact point of $A\subset X$ , if and only if , there is $\sigma \in F$ such that $x \in S$ and $S \cap A \neq\emptyset$ for all $S \in\sigma$ .
When is there an $F$ compatible with the topology on X in which finite subsets of X and X are closed ?
contact point is also known as adherent point.
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.
2023 Miklós Schweitzer, 5
Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.
2018 AIME Problems, 2
Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.
2012 Today's Calculation Of Integral, 826
Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.
2021 CIIM, 3
Let $m,n$ and $N$ be positive integers and $\mathbb{Z}_{N}=\{0,1,\dots,N-1\}$ a set of residues modulo $N$. Consider a table $m\times n$ such that each one of the $mn$ cells has an element of $\mathbb{Z}_{N}$. A [i]move[/i] is choose an element $g\in \mathbb{Z}_{N}$, a cell in the table and add $+g$ to the elements in the same row/column of the chosen cell(the sum is modulo $N$). Prove that if $N$ is coprime with $m-1,n-1,m+n-1$ then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.
2004 All-Russian Olympiad, 3
On a table there are 2004 boxes, and in each box a ball lies. I know that some the balls are white and that the number of white balls is even. In each case I may point to two arbitrary boxes and ask whether in the box contains at least a white ball lies. After which minimum number of questions I can indicate two boxes for sure, in which white balls lie?
2024 India IMOTC, 4
Let $n$ be a positive integer. Let $s: \mathbb N \to \{1, \ldots, n\}$ be a function such that $n$ divides $m-s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \ldots$ be a sequence such that $a_0=0$ and \[a_{k}=a_{k-1}+s(k) \text{ for all }k\ge 1.\]
Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$.
[i]Proposed by N.V. Tejaswi[/i]
2017 District Olympiad, 2
Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.
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.
2004 Romania National Olympiad, 4
Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$.
(a) Prove that $-1$ is the square of an element from $\mathcal K.$
(b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$.
(c) Can $0$ be written in the same way?
[i]Marian Andronache[/i]
1975 Putnam, B1
Consider the additive group $\mathbb{Z}^{2}$. Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$.
Let $H_{xy}$ be the smallest subgroup containing $(0,x)$ and $(1,y)$. Find some pair $(x,y)$ with $x>0$ such that $H=H_{xy}$.
1977 IMO Longlists, 34
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 Kazakhstan National Olympiad, 6
Let $P(x)$ be polynomial with integer coefficients.
Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$
1972 Miklós Schweitzer, 4
Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite.
[i]J. Pelikan[/i]