Olympiad problems

2007 IMC, 5

Tags: inequalities , induction , vector , abstract algebra , invariant , algebra , polynomial

For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold: (1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$, (2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and (3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.

2004 Alexandru Myller, 1

Tags: abstract algebra , field theory , ring theory , freshman s dream , number theory

Show that the equation $ (x+y)^{-1}=x^{-1}+y^{-1} $ has a solution in the field of integers modulo $ p $ if and only if $ p $ is a prime congruent to $ 1 $ modulo $ 3. $ [i]Mihai Piticari[/i]

1954 Miklós Schweitzer, 7

Tags: group theory , abstract algebra

[b]7.[/b] Find the finite groups having only one proper maximal subgroup. [b](A.12)[/b]

2006 VJIMC, Problem 2

Tags: group theory , abstract algebra

Let $(G,\cdot)$ be a finite group of order $n$. Show that each element of $G$ is a square if and only if $n$ is odd.

1984 Putnam, B3

Tags: abstract algebra , set

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$, $(\text i)$ $x*z=y*z$ implies $x=y$ $(\text{ii})$ $x*(y*z)\ne(x*y)*z$

2008 IMC, 5

Tags: group theory , abstract algebra

Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\text{Aut } H| > |\text{Aut } G|$? Disprove or provide an example. Here the notation $ |\text{Aut } X|$ for some group $ X$ denotes the number of isomorphisms from $ X$ to itself.

2001 District Olympiad, 1

Tags: group theory , abstract algebra , pigeonhole principle , superior algebra

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]

1995 Irish Math Olympiad, 2

Tags: abstract algebra , trigonometry , algebra

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$.

1984 Canada National Olympiad, 1

Tags: modular arithmetic , abstract algebra , number theory

Prove that the sum of the squares of $1984$ consecutive positive integers cannot be the square of an integer.

1979 Miklós Schweitzer, 4

Tags: group theory , abstract algebra , geometry , 3d geometry , sphere , superior algebra

For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.) [i]Z. Szabo[/i]

2008 District Olympiad, 3

Tags: group theory , abstract algebra , superior algebra

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$.

2005 Today's Calculation Of Integral, 44

Tags: calculus , integration , trigonometry , induction , abstract algebra , calculus computations

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

2020 Jozsef Wildt International Math Competition, W17

Tags: field , group theory , field theory , abstract algebra

Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that: $$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$ [i]Proposed by Ovidiu Pop[/i]

2006 VJIMC, Problem 1

Tags: ring theory , abstract algebra

(a) Let $u$ and $v$ be two nilpotent elements in a commutative ring (with or without unity). Prove that $u+v$ is also nilpotent. (b) Show an example of a (non-commutative) ring $R$ and nilpotent elements $u,v\in R$ such that $u+v$ is not nilpotent.

1975 Putnam, B1

Tags: group theory , abstract algebra

Consider the additive group $\mathbb{Z}^{2}$. Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$. Let $H_{xy}$ be the smallest subgroup containing $(0,x)$ and $(1,y)$. Find some pair $(x,y)$ with $x>0$ such that $H=H_{xy}$.

2010 Canada National Olympiad, 4

Tags: abstract algebra , induction , combinatorics

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.

2005 Romania National Olympiad, 2

Tags: group theory , abstract algebra , function , superior algebra

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]

2010 Romania National Olympiad, 3

Tags: inequalities , group theory , abstract algebra , superior algebra

Let $G$ be a finite group of order $n$. Define the set \[H=\{x:x\in G\text{ and }x^2=e\},\] where $e$ is the neutral element of $G$. Let $p=|H|$ be the cardinality of $H$. Prove that a) $|H\cap xH|\ge 2p-n$, for any $x\in G$, where $xH=\{xh:h\in H\}$. b) If $p>\frac{3n}{4}$, then $G$ is commutative. c) If $\frac{n}{2}<p\le\frac{3n}{4}$, then $G$ is non-commutative. [i]Marian Andronache[/i]

2011 Romania National Olympiad, 1

Tags: abstract algebra , ring theory

Prove that a ring that has a prime characteristic admits nonzero nilpotent elements if and only if its characteristic divides the number of its units.

2007 Bulgaria National Olympiad, 3

Tags: algebra , polynomial , induction , calculus , integration , abstract algebra , geometry

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 $P(x)=Q(x)^2$.

2012 IMC, 3

Tags: group theory , abstract algebra

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]

2011 Pre - Vietnam Mathematical Olympiad, 1

Tags: abstract algebra , number theory

Determine all values of $n$ satisfied the following condition: there's exist a cyclic $(a_1,a_2,a_3,...,a_n)$ of $(1,2,3,...,n)$ such that $\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}$ is a complete residue systems modulo $n$.

2012 Centers of Excellency of Suceava, 1

Tags: group theory , abstract algebra

Let be a natural number $ n\ge 2, $ a group $ G $ and two elements of it $ e_1,e_2 $ such that $ e_2e_1x=xe_2e_1, $ for any element $ x $ of $ G. $ Prove that $ \left( e_1xe_2 \right)^n =e_1x^ne_2, $ for any element $ x $ of $ G, $ if and only if $ e_2e_1=\left( e_2e_1\right)^n. $ [i]Ion Bursuc[/i]

2002 Iran Team Selection Test, 10

Tags: geometry , rectangle , linear algebra , matrix , vector , abstract algebra , function

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

2012 China Team Selection Test, 3

Tags: function , group theory , abstract algebra , floor function , induction , algebra

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.