Found problems: 339
2017 Romania Team Selection Test, P1
a) Determine all 4-tuples $(x_0,x_1,x_2,x_3)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 4) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}$ is an interger.
b)Show that there are infinitely many 5-tuples $(x_0,x_1,x_2,x_3,x_4)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 5) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0}$ is an interger.
2013 Miklós Schweitzer, 5
A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.
[i]Proposed by Péter Nagy Tibor[/i]
PEN M Problems, 27
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.
2006 Pre-Preparation Course Examination, 4
If $d\in \mathbb{Q}$, is there always an $\omega \in \mathbb{C}$ such that $\omega ^n=1$ for some $n\in \mathbb{N}$ and $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\omega)$?
2005 IMC, 6
6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.
PEN D Problems, 2
Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]
2005 Olympic Revenge, 4
Let A be a symmetric matrix such that the sum of elements of any row is zero.
Show that all elements in the main diagonal of cofator matrix of A are equal.
2012 Bulgaria National Olympiad, 1
Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.
2021 Science ON all problems, 2
Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\
$\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\
$\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\
$\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true?
\\ \\
[i] (Bogdan Blaga)[/i]
1992 Miklós Schweitzer, 3
Call a (non-trivial) lattice class a pseudo-variety if it is closed under taking a homomorphic image, a direct product, and a convex subset. Prove that the smallest distributive pseudo-variety cannot be defined by a first-order set of formulas.
1959 Miklós Schweitzer, 3
[b]3.[/b]Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$. Show that if, for any $k$, the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$, then every $H_k (k=1, \dots , n)$ is of finite index in $G$. [b](A. 15)[/b]
2008 Miklós Schweitzer, 4
Let $A$ be a subgroup of the symmetric group $S_n$, and $G$ be a normal subgroup of $A$. Show that if $G$ is transitive, then $|A\colon G|\le 5^{n-1}$
(translated by Miklós Maróti)
2019 LIMIT Category B, Problem 3
A subset $W$ of the set of real numbers is called a ring if it contains $1$ and if for all $a,b\in W$, the numbers $a-b$ and $ab$ are also in $W$. Let $S=\left\{\frac m{2^n}|m,n\in\mathbb Z\right\}$ and $T=\left\{\frac pq|p,q\in\mathbb Z,q\text{ odd}\right\}$. Then
$\textbf{(A)}~\text{neither }S\text{ nor }T\text{ is a ring}$
$\textbf{(B)}~S\text{ is a ring, }T\text{ is not a ring}$
$\textbf{(C)}~T\text{ is a ring, }S\text{ is not a ring}$
$\textbf{(D)}~\text{both }S\text{ and }T\text{ are rings}$
2005 IMO Shortlist, 7
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
1982 IMO Longlists, 18
You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that
\[(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},\]
where $x^{-1}$ is the element for which $x^{-1}x = xx^{-1} = e$, where $e$ is the element of the system such that for all $a$ the equality $ea = ae = a$ holds.
2014 IMS, 5
Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomorphisms from $G_2$ to $H$. Prove that $G_1$ and $G_2$ are Isomorphic.
1995 Irish Math Olympiad, 2
Let $ a,b,c$ be complex numbers. Prove that if all the roots of the equation $ x^3\plus{}ax^2\plus{}bx\plus{}c\equal{}0$ are of module $ 1$, then so are the roots of the equation $ x^3\plus{}|a|x^2\plus{}|b|x\plus{}|c|\equal{}0$.
2018 Miklós Schweitzer, 6
Prove that if $a$ is an integer and $d$ is a positive divisor of the number $a^4+a^3+2a^2-4a+3$, then $d$ is a fourth power modulo $13$.
2010 District Olympiad, 3
Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.
2005 Iran MO (3rd Round), 2
We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i,
A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$.
Say the the following sets have the relation $\sim$ or not ?
a) Natural numbers and composite numbers.
b) Rational numbers and rational numbers with finite digits in base 10.
c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$
d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$
2011 Croatia Team Selection Test, 2
There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).
PEN J Problems, 2
Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]
2018 Saint Petersburg Mathematical Olympiad, 4
$f(x)$ is polynomial with integer coefficients, with module not exceeded $5*10^6$. $f(x)=nx$ has integer root for $n=1,2,...,20$. Prove that $f(0)=0$
2005 Gheorghe Vranceanu, 1
For a natural number $ n\ge 2, $ prove that the $ \text{n-ary} $ direct product of the group of order $ 2 $ is abelian and isomorphic with the group of the power set of a set under symmetric difference.
2001 District Olympiad, 1
For any $n\in \mathbb{N}^*$, let $H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}$.
a) Prove that $H_n$ is a subgroup of the group $(Q,+)$ and that $Q=\bigcup_{n\in \mathbb{N}^*} H_n$;
b) Prove that if $G_1,G_2,\ldots, G_m$ are subgroups of the group $(Q,+)$ and $G_i\neq Q,\ (\forall) 1\le i\le m$, then
$G_1\cup G_2\cup \ldots \cup G_m\neq Q$
[i]Marian Andronache & Ion Savu[/i]