Olympiad problems

2012-2013 SDML (Middle School), 5

Tags: abstract algebra

What is the hundreds digit of the sum below? $$1+12+123+1234+12345+123456+1234567+12345678+123456789$$ $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2004 Romania National Olympiad, 4

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

Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$. (a) Prove that $-1$ is the square of an element from $\mathcal K.$ (b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$. (c) Can $0$ be written in the same way? [i]Marian Andronache[/i]

2003 SNSB Admission, 2

Tags: algebra , polynomial , abstract algebra , galois theory

Let be the polynomial $ f=X^4+X^2\in\mathbb{Z}_2[X] $ Find: a) its degree.. b) the splitting field of $ f $ c) the Galois group of $ f $ (Galois group of its splitting field)

1973 Miklós Schweitzer, 1

Tags: group theory , abstract algebra , linear algebra , matrix , modular arithmetic , galois theory , superior algebra

We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$. [i]J. Erdos[/i]

2020 Putnam, B1

Tags: abstract algebra

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 $2020$.

1985 Traian Lălescu, 1.3

Tags: group theory , abstract algebra , superior algebra

Let $ G $ be a finite group of odd order having, at least, three elements. For $ a\in G $ denote $ n(a) $ as the number of ways $ a $ can be written as a product of two distinct elements of $ G. $ Prove that $ \sum_{\substack{a\in G\\a\neq\text{id}}} n(a) $ is a perfect square.

2008 USAMO, 6

Tags: induction , abstract algebra , vector , group theory

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).

1987 Traian Lălescu, 2.3

Tags: group theory , abstract algebra , centralizer , normalizer

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. $

2022 Moldova EGMO TST, 4

Tags: algebra , polynomial , abstract algebra , modular arithmetic

Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$. for all $n \geq 0$.

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.

2011 Northern Summer Camp Of Mathematics, 4

Tags: abstract algebra

Find all positive integers $n$ such that $7^n+147$ is a perfect square.

2013 Miklós Schweitzer, 5

Tags: abstract algebra , advanced fields

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]

2016 IMC, 3

Tags: abstract algebra , function

Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties: (i) $f(x)\neq x$, (ii) $f(f(x))=x$, (iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$. Prove that $n\equiv 2 \pmod4$. (Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)

PEN D Problems, 12

Tags: congruence , modular arithmetic , floor function , abstract algebra , number theory , relatively prime , group theory

Suppose that $m>2$, and let $P$ be the product of the positive integers less than $m$ that are relatively prime to $m$. Show that $P \equiv -1 \pmod{m}$ if $m=4$, $p^n$, or $2p^{n}$, where $p$ is an odd prime, and $P \equiv 1 \pmod{m}$ otherwise.

2001 Miklós Schweitzer, 4

Tags: abstract algebra , units , ring theory

Find the units of $R=\mathbb Z[t][\sqrt{t^2-1}]$.

2024 IMC, 4

Tags: group theory , abstract algebra , combinatorics of words , subgroup , sequence

Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).

2009 National Olympiad First Round, 32

Tags: algorithm , abstract algebra , group theory

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) $) $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None}$

2010 Iran MO (3rd Round), 7

Tags: function , abstract algebra , algebra

[b]interesting function[/b] $S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and $f : P(S) \rightarrow \mathbb N$ is a function with these properties: for every subset $A$ of $S$ we have $f(A)=f(S-A)$. for every two subsets of $S$ like $A$ and $B$ we have $max(f(A),f(B))\ge f(A\cup B)$ prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$. time allowed for this question was 1 hours and 30 minutes.

1994 AMC 12/AHSME, 27

Tags: abstract algebra , probability

A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} $

2010 Laurențiu Panaitopol, Tulcea, 4

Tags: abstract algebra , ring theory , polynomial , polynomial ring

Let be a ring $ R $ which has the property that there exist two distinct natural numbers $ s,t $ such that for any element $ x $ of $ R, $ the equation $ x^s=x^t $ is true. Show that there exists a polynom in $ R[X] $ of degree $$ |s-t|\left( 1+|s-t| \right) $$ such that all the elements of $ R $ are roots of it.

2025 District Olympiad, P1

Tags: group theory , abstract algebra

Let $G$ be a group and $A$ a nonempty subset of $G$. Let $AA=\{xy\mid x,y\in A\}$. [list=a] [*] Prove that if $G$ is finite, then $AA=A$ if and only if $|A|=|AA|$ and $e\in A$. [*] Give an example of a group $G$ and a nonempty subset $A$ of $G$ such that $AA\neq A$, $|AA|=|A|$ and $AA$ is a proper subgroup of $G$. [/list] [i]Mathematical Gazette - Robert Rogozsan[/i]

1979 Miklós Schweitzer, 4

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

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]

2001 District Olympiad, 1

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

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]

2018 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , polynomial , abstract algebra

$f(x)$ 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 $f(0)=0$

1993 Hungary-Israel Binational, 5

Tags: group theory , abstract algebra , superior algebra

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 $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?