Found problems: 250
2008 IMC, 4
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?
1977 Miklós Schweitzer, 7
Let $ G$ be a locally compact solvable group, let $ c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $ f$ and $ g$ on $ G$ satisfy \[ \sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .\] Prove that if $ f$ is a bounded function and \[ \inf_{x \in G} \textrm{Re} f(x) \chi(x) >0\] for some continuous (complex) character $ \chi$ of $ G$, then $ g$ is continuous.
[i]L. Szekelyhidi[/i]
2012 France Team Selection Test, 1
Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$).
1) If $k=2n+1$, prove that there exists a person who knows all others.
2) If $k=2n+2$, give an example of such a group in which no-one knows all others.
2011 Indonesia TST, 2
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$).
2009 IMS, 1
$ G$ is a group. Prove that the following are equivalent:
1. All subgroups of $ G$ are normal.
2. For all $ a,b\in G$ there is an integer $ m$ such that $ (ab)^m\equal{}ba$.
2003 Romania National Olympiad, 4
$ i(L) $ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $ L $ such that the set of elements of $ L, $ along with these additive and multiplicative operations, form a ring. Prove that
[b]a)[/b] $ i\left( \mathbb{Z}_{12} \right) =4. $
[b]b)[/b] $ i(A\times B)\ge i(A)i(B) , $ for any two finite commutative groups $ B $ and $ A. $
[b]c)[/b] there exist two sequences $ \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} $ of finite commutative groups such that
$$ \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 $$
and
$$ \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty. $$
[i]Barbu Berceanu[/i]
2012 India National Olympiad, 6
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and
$(i) f(xy) + f(x)f(y) = f(x) + f(y)$
$(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $
for all $x,y \in \mathbb{Z}$, simultaneously.
$(a)$ Find the set of all possible values of the function $f$.
$(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.
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]
1986 Traian Lălescu, 1.2
Let $ K $ be the group of Klein. Prove that:
[b]a)[/b] There is an unique division ring (up to isomorphism), $ D, $ such that $ (D,+)\cong K. $
[b]b)[/b] There are no division rings $ A $ such that $ (A\setminus\{ 0\} ,+)\cong K. $
1996 Turkey MO (2nd round), 2
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
2014 USA Team Selection Test, 3
For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if
$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and
$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.
[i]Proposed by Noga Alon and Jean Bourgain[/i]
2011 District Olympiad, 2
Let $ G $ be the set of matrices of the form $ \begin{pmatrix} a&b\\0&1 \end{pmatrix} , $ with $ a,b\in\mathbb{Z}_7,a\neq 0. $
[b]a)[/b] Verify that $ G $ is a group.
[b]b)[/b] Show that $ \text{Hom}\left( (G,\cdot) ; \left( \mathbb{Z}_7,+ \right) \right) =\{ 0\} $
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.
1972 Miklós Schweitzer, 4
Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite.
[i]J. Pelikan[/i]
2007 Romania National Olympiad, 4
Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$, generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$, denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$.
Show that for any $k$, $|S(k)|\leq n-1$.
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.
2004 Iran MO (3rd Round), 17
Let $ p\equal{}4k\plus{}1$ be a prime. Prove that $ p$ has at least $ \frac{\phi(p\minus{}1)}2$ primitive roots.
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.
2004 Bulgaria National Olympiad, 4
In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.
2010 Gheorghe Vranceanu, 1
Let be a semigroup with the property that for any two elements of it $ a,b, $ there is another element $ c $ such that $ axa=b. $ Prove that it's a group.
2011 District Olympiad, 4
Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove:
[b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $
[b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $
1987 Greece National Olympiad, 1
a) Prove that every sub-group $(A,+)$ of group $(\mathbb{Z},+)$ is in the form $A=n \cdot \mathbb{Z}$ for some $n \in \mathbb{Z}$ where $n \cdot \mathbb{Z}=\{n \cdot x/x\in\mathbb{Z}\}$.
b) Using problem (a) , prove that the greatest common divisor $d$ of non zero integers $a_1, a_2,... ,a_n$ is given by relation $d=\lambda_1a_1+\lambda_2 a_2+...\lambda_n a_n$ with $\lambda_i\in\mathbb{Z}, \,\, i=1,2,...,n$
1989 Putnam, B2
Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group?
I haven't had much group theory at this point...
1951 Miklós Schweitzer, 14
For which commutative finite groups is the product of all elements equal to the unit element?
2014 Romania National Olympiad, 4
Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $
[b]a)[/b] Prove that the order of $ G $ is a power of $ p. $
[b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $