Found problems: 339
2014 IMS, 3
Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.
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.
2004 Gheorghe Vranceanu, 1
Let $(G,\cdot)$ be a group, and let $H_1,H_2$ be proper subgroups s.t. $H_1\cap H_2=\{e\}$, where $e$ is the identity element of $G$. They also have the following properties:
[b]i)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_1\setminus\{e\}\Rightarrow xy\in H_2$
[b]ii)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_2\setminus\{e\}\Rightarrow xy\in H_1$
Prove that:
[b]a)[/b] $|H_1|=|H_2|$
[b]b)[/b] $|G|=|H_1|\cdot |H_2|$
2022-2023 OMMC, 22
Find the number of ordered pairs of integers $(x, y)$ with $0 \le x, y \le 40$ where $$\frac{x^2-xy^2+1}{41}$$ is an integer.
2012 IMC, 3
Given an integer $n>1$, let $S_n$ be the group of permutations of the numbers $1,\;2,\;3,\;\ldots,\;n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
1985 Iran MO (2nd round), 4
Let $G$ be a group and let $a$ be a constant member of it. Prove that
\[G_a = \{x | \exists n \in \mathbb Z , x=a^n\}\]
Is a subgroup of $G.$
1990 Turkey Team Selection Test, 6
Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.
2007 Miklós Schweitzer, 4
Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$).
(translated by Miklós Maróti)
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.
2008 District Olympiad, 3
Let $ A$ be a commutative unitary ring with an odd number of elements.
Prove that the number of solutions of the equation $ x^2 \equal{} x$ (in $ A$) divides the number of invertible elements of $ A$.
2017 Danube Mathematical Olympiad, 4
Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.
2007 Pre-Preparation Course Examination, 2
Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.
2020 Dürer Math Competition (First Round), P5
Let $p$ be prime and $ k > 1$ be a divisor of $p-1$. Show that if a polynomial of degree $k$ with integer coefficients attains every possible value modulo $ p$ that is $(0,1,\dots, p-1)$ at integer inputs then its leading coefficient must be divisible by $p$.
[hide=Note]Note: the leading coefficient of a polynomial of degree d is the coefficient of the $x_d$ term.[/hide]
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.
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.
2014 IMS, 7
Let $G$ be a finite group such that for every two subgroups of it like $H$ and $K$, $H \cong K$ or $H \subseteq K$ or $K \subseteq H$. Prove that we can produce each subgroup of $G$ with 2 elements at most.
PEN D Problems, 2
Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]
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.
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)
2014 Contests, 3
Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.
2024 Romania National Olympiad, 4
Let $\mathbb{L}$ be a finite field with $q$ elements. Prove that:
a) If $q \equiv 3 \pmod 4$ and $n \ge 2$ is a positive integer divisible by $q-1,$ then $x^n=(x^2+1)^n$ for all $x \in \mathbb{L}^{\times}.$
b) If there exists a positive integer $n \ge 2$ such that $x^n=(x^2+1)^n$ for all $x \in \mathbb{L}^{\times},$ then $q \equiv 3 \pmod 4$ and $q-1$ divides $n.$
1998 Romania National Olympiad, 2
$\textbf{a) }$ Let $p \geq 2$ be a natural number and $G_p = \bigcup\limits_{n \in \mathbb{N}} \lbrace z \in \mathbb{C} \mid z^{p^n}=1 \rbrace.$ Prove that $(G_p, \cdot)$ is a subgroup of $(\mathbb{C}^*, \cdot).$
$\textbf{b) }$ Let $(H, \cdot)$ be an infinite subgroup of $(\mathbb{C}^*, \cdot).$ Prove that all proper subgroups of $H$ are finite if and only if $H=G_p$ for some prime $p.$
2014 Miklós Schweitzer, 6
Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.
2023 District Olympiad, P2
Let $(G,\cdot)$ be a grup with neutral element $e{}$, and let $H{}$ and $K$ be proper subgroups of $G$, satisfying $H\cap K=\{e\}$. It is known that $(G\setminus(H\cup K))\cup\{e\}$ is closed under the operation of $G$. Prove that $x^2=e$ for all the elements $x{}$ of $G{}$.
2019 IMAR Test, 3
Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that:
$$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$