Found problems: 339
2006 Mathematics for Its Sake, 1
Determine the number of polynomials of degree $ 3 $ that are irreducible over the field of integers modulo a prime.
2011 Miklós Schweitzer, 4
Let G, H be two finite groups, and let $\varphi, \psi: G \to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the identity element of G but whose images under $\varphi$ and $\psi$ are the same.
1964 Miklós Schweitzer, 3
Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.
2009 Indonesia TST, 3
Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.
1978 Germany Team Selection Test, 4
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.
1999 Romania National Olympiad, 2
For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it.
Decide whether the following statements are true or false.
a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$
b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$
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?
2008 District Olympiad, 1
Let $ z \in \mathbb{C}$ such that for all $ k \in \overline{1, 3}$, $ |z^k \plus{} 1| \le 1$. Prove that $ z \equal{} 0$.
2010 Iran MO (3rd Round), 2
prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)
2010 N.N. Mihăileanu Individual, 4
If $ p $ is an odd prime, then the following characterization holds.
$$ 2^{p-1}\equiv 1\pmod{p^2}\iff \sum_{2=q}^{(p-1)/2} q^{p-2}\equiv -1\pmod p $$
[i]Marius Cavachi[/i]
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.
2007 Bulgaria National Olympiad, 3
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
2019 LIMIT Category C, Problem 6
Which of the following are true?
$\textbf{(A)}~GL(n,\mathbb R)\text{ is connected}$
$\textbf{(B)}~GL(n,\mathbb C)\text{ is connected}$
$\textbf{(C)}~O(n,\mathbb R)\text{ is connected}$
$\textbf{(D)}~O(n,\mathbb C)\text{ is connected}$
2024 Romania National Olympiad, 2
Let $(\mathbb{K},+, \cdot)$ be a division ring in which $x^2y=yx^2,$ for all $x,y \in \mathbb{K}.$ Prove that $(\mathbb{K},+, \cdot)$ is commutative.
2010 Contests, 4
Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a
sequence of operations of this type?
Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.
2021 Science ON all problems, 1
Are there any integers $a,b$ and $c$, not all of them $0$, such that
$$a^2=2021b^2+2022c^2~~?$$
[i] (Cosmin Gavrilă)[/i]
1954 Miklós Schweitzer, 7
[b]7.[/b] Find the finite groups having only one proper maximal subgroup. [b](A.12)[/b]
2007 IberoAmerican, 5
Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.
2005 Romania National Olympiad, 2
Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$).
a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$;
b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$.
[i]Calin Popescu[/i]
1996 Miklós Schweitzer, 4
Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.
PEN R Problems, 3
Prove no three lattice points in the plane form an equilateral triangle.
1994 USAMO, 2
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, $\,\ldots, \,$ red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides
are red, blue, red, blue, red, blue, $\, \ldots, \,$ red, yellow, blue?
1987 Traian Lălescu, 2.3
Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $
[i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $
2002 VJIMC, Problem 2
A ring $R$ (not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that $R$ is finite.
2021 Science ON grade VIII, 1
Are there any integers $a,b$ and $c$, not all of them $0$, such that
$$a^2=2021b^2+2022c^2~~?$$
[i] (Cosmin Gavrilă)[/i]