Found problems: 339
2005 Vietnam Team Selection Test, 2
Let $p\in \mathbb P,p>3$. Calcute:
a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$
b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$
1993 Hungary-Israel Binational, 3
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.
Show that every element of $S_{n}$ is a product of $2$-cycles.
2019 District Olympiad, 1
Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$
$\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism.
$\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?
1986 Traian Lălescu, 1.1
Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties:
$ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $
$ \text{(ii)}\quad \vartheta (1)=1 $
$ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $
[b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism.
[b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.
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]
2006 Iran MO (3rd Round), 3
$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.
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.
2014 USA Team Selection Test, 3
For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if
$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and
$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.
[i]Proposed by Noga Alon and Jean Bourgain[/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]
1996 IMC, 9
Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$, where
$$A=\begin{pmatrix}
2 &0\\
0&1
\end{pmatrix},\;
B=\begin{pmatrix}
1 &1\\
0&1
\end{pmatrix}$$.
Let $H$ consist of the matrices $\begin{pmatrix}
a_{11} &a_{12}\\
a_{21}& a_{22}
\end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$.
a) Show that $H$ is an abelian subgroup of $G$.
b) Show that $H$ is not finitely generated.
2019 LIMIT Category C, Problem 6
Which of the following are true?
$\textbf{(A)}~GL(n,\mathbb R)\text{ is connected}$
$\textbf{(B)}~GL(n,\mathbb C)\text{ is connected}$
$\textbf{(C)}~O(n,\mathbb R)\text{ is connected}$
$\textbf{(D)}~O(n,\mathbb C)\text{ is connected}$
2015 District Olympiad, 4
Let $ m $ be a non-negative ineger, $ n\ge 2 $ be a natural number, $ A $ be a ring which has exactly $ n $ elements, and an element $ a $ of $ A $ such that $ 1-a^k $ is invertible, for all $ k\in\{ m+1,m+2,...,m+n-1\} . $
Prove that $ a $ is nilpotent.
2016 CHMMC (Fall), 13
A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?
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 $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
1987 Greece National Olympiad, 1
a) Prove that every sub-group $(A,+)$ of group $(\mathbb{Z},+)$ is in the form $A=n \cdot \mathbb{Z}$ for some $n \in \mathbb{Z}$ where $n \cdot \mathbb{Z}=\{n \cdot x/x\in\mathbb{Z}\}$.
b) Using problem (a) , prove that the greatest common divisor $d$ of non zero integers $a_1, a_2,... ,a_n$ is given by relation $d=\lambda_1a_1+\lambda_2 a_2+...\lambda_n a_n$ with $\lambda_i\in\mathbb{Z}, \,\, i=1,2,...,n$
2010 Iran MO (3rd Round), 4
a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms:
i-$\{0\}$.
ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$.
iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$)
b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)
2006 Iran MO (3rd Round), 5
For each $n$, define $L(n)$ to be the number of natural numbers $1\leq a\leq n$ such that $n\mid a^{n}-1$. If $p_{1},p_{2},\ldots,p_{k}$ are the prime divisors of $n$, define $T(n)$ as $(p_{1}-1)(p_{2}-1)\cdots(p_{k}-1)$.
a) Prove that for each $n\in\mathbb N$ we have $n\mid L(n)T(n)$.
b) Prove that if $\gcd(n,T(n))=1$ then $\varphi(n) | L(n)T(n)$.
2004 Silk Road, 4
Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar.
Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.
PEN H Problems, 49
Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.
2009 Putnam, A5
Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$
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]
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!}.$
1986 Miklós Schweitzer, 5
Prove that existence of a constant $c$ with the following property: for every composite integer $n$, there exists a group whose order is divisible by $n$ and is less than $n^c$, and that contains no element of order $n$. [P. P. Palfy]
2018 PUMaC Team Round, 16
Let $N$ be the number of subsets $B$ of the set $\{1,2,\dots,2018\}$ such that the sum of the elements of $B$ is congruent to $2018$ modulo $2048$. Find the remainder when $N$ is divided by $1000$.
2015 Harvard-MIT Mathematics Tournament, 8
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
[list]
[*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
[*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
[/list]