Found problems: 339
2017 CIIM, Problem 3
Let $G$ be a finite abelian group and $f :\mathbb{Z}^+ \to G$ a completely multiplicative function (i.e. $f(mn) = f(m)f(n)$ for any positive integers $m, n$). Prove that there are infinitely many positive integers $k$ such that $f(k) = f(k + 1).$
1978 Miklós Schweitzer, 2
For a distributive lattice $ L$, consider the following two statements:
(A) Every ideal of $ L$ is the kernel of at least two different homomorphisms.
(B) $ L$ contains no maximal ideal.
Which one of these statements implies the other?
(Every homomorphism $ \varphi$ of $ L$ induces an equivalence relation on $ L$: $ a \sim b$ if and only if $ a \varphi\equal{}b \varphi$. We do not consider two homomorphisms different if they imply the same equivalence relation.)
[i]J. Varlet, E. Fried[/i]
2013 Miklós Schweitzer, 3
Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$-Sylow subgroup of $A_n$.
[i]Proposed by Péter Pál Pálfy[/i]
2020 Putnam, B1
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3$). Let
\[
S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.
\]
Determine $S$ modulo $2020$.
2019 LIMIT Category C, Problem 5
Let $G=(S^1,\cdot)$ be a group. Then its nontrivial subgroups
$\textbf{(A)}~\text{are necessarily finite}$
$\textbf{(B)}~\text{can be infinite}$
$\textbf{(C)}~\text{can be dense in }S^1$
$\textbf{(D)}~\text{None of the above}$
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?
2013 Albania Team Selection Test, 5
Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that :
$(2^k)!=2^n*m$
2024 India National Olympiad, 5
Let points $A_1$, $A_2$ and $A_3$ lie on the circle $\Gamma$ in a counter-clockwise order, and let $P$ be a point in the same plane. For $i \in \{1,2,3\}$, let $\tau_i$ denote the counter-clockwise rotation of the plane centred at $A_i$, where the angle of rotation is equial to the angle at vertex $A_i$ in $\triangle A_1A_2A_3$. Further, define $P_i$ to be the point $\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))$, where the indices are taken modulo $3$ (i.e., $\tau_4 = \tau_1$ and $\tau_5 = \tau_2$).
Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$.
[i]Proposed by Anant Mudgal[/i]
1993 Hungary-Israel Binational, 6
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 $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$
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.
2005 MOP Homework, 1
We call a natural number 3-partite if the set of its divisors can be partitioned into 3 subsets each with the same sum. Show that there exist infinitely many 3-partite numbers.
1987 Traian Lălescu, 2.1
Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.
Gheorghe Țițeica 2024, P2
a) Let $n$ be a positive integer $G$ be a a group with $|G|<\frac{4n^2}{n-\varphi(n)}$. Suppose that $Z(G)$ contains at least $\varphi(n)+1$ elements of order $n$. Prove that $G$ is abelian.
b) Find a noncommutative group $G$ with $16$ elements such that $Z(G)$ contains two elements of order two.
[i]Robert Rogozsan & Filip Munteanu[/i]
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.
2009 National Olympiad First Round, 32
There are $ n$ sets having $ 4$ elements each. The difference set of any two of the sets is equal to one of the $ n$ sets. $ n$ can be at most ? (A difference set of $A$ and $B$ is $ (A\setminus B)\cup(B\setminus A) $)
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None}$
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.
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$.
2007 Nicolae Coculescu, 3
Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition:
$$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$
[i]Marius Perianu[/i]
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.
1998 Romania National Olympiad, 3
A ring $A$ is called Boolean if $x^2 = x$ for every $x \in A$. Prove that:
a) A finite set $A$ with $n \geq 2$ elements can be equipped with the structure of a Boolean ring if and only if $n = 2^k$ for some positive integer $k$.
b) The set of nonnegative integers can be equipped with the structure of a Boolean ring.
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.
2019 LIMIT Category C, Problem 3
$G$ be a group and $H\le G$. Then which of the following are true?
$\textbf{(A)}~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H$
$\textbf{(B)}~\exists G,H\text{ and }H\le G\text{ with }H\cong G$
$\textbf{(C)}~\text{All subgroups are normal, then }G\text{ is abelian.}$
$\textbf{(D)}~\text{None of the above}$
2009 Romania National Olympiad, 3
Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares.
2011 Romania National Olympiad, 3
The equation $ x^{n+1} +x=0 $ admits $ 0 $ and $ 1 $ as its unique solutions in a ring of order $ n\ge 2. $
Prove that this ring is a skew field.
2013 Iran Team Selection Test, 2
Find the maximum number of subsets from $\left \{ 1,...,n \right \}$ such that for any two of them like $A,B$ if $A\subset B$ then $\left | B-A \right |\geq 3$. (Here $\left | X \right |$ is the number of elements of the set $X$.)