Found problems: 250
1968 Putnam, B2
Let $G$ be a finite group with $n$ elements and $K$ a subset of $G$ with more than $\frac{n}{2}$ elements. Show that for any $g\in G$ one can find $h,k\in K$ such that $g=h\cdot k$.
PEN R Problems, 3
Prove no three lattice points in the plane form an equilateral triangle.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3
Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.
2009 Miklós Schweitzer, 5
Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then
(i) $ G$ is not simple;
(ii) $ G$ contains a normal subgroup of order $ m$.
1994 USAMO, 2
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, $\,\ldots, \,$ red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides
are red, blue, red, blue, red, blue, $\, \ldots, \,$ red, yellow, blue?
2019 LIMIT Category C, Problem 3
$G$ be a group and $H\le G$. Then which of the following are true?
$\textbf{(A)}~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H$
$\textbf{(B)}~\exists G,H\text{ and }H\le G\text{ with }H\cong G$
$\textbf{(C)}~\text{All subgroups are normal, then }G\text{ is abelian.}$
$\textbf{(D)}~\text{None of the above}$
2002 Iran Team Selection Test, 2
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2007 Purple Comet Problems, 15
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.
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).
1997 Miklós Schweitzer, 7
Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality.
$$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2$$
Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$.
2014 Cezar Ivănescu, 1
Let $ S $ be a nonempty subset of a finite group $ G, $ and $ \left( S^j \right)_{j\ge 1} $ be a sequence of sets defined as $ S^j=\left.\left\{\underbrace{xy\cdots z}_{\text{j terms}} \right| \underbrace{x,y,\cdots ,z}_{\text{j terms}} \in S \right\} . $ Prove that:
[b]a)[/b] $ \exists i_0\in\mathbb{N}^*\quad i\ge i_0\implies \left| S^i\right| =\left| S^{1+i}\right| $
[b]b)[/b] $ S^{|G|}\le G $
2006 Romania National Olympiad, 1
Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent:
(a) $\displaystyle 1+1=0$;
(b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.
2006 Germany Team Selection Test, 3
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]
2012 Gheorghe Vranceanu, 2
A group $ G $ of order at least $ 4 $ has the property that there exists a natural number $ n\not\in\{ 1,|G| \} $ such that $ G $ admits exactly $ \binom{|G|-1}{n-1} $ subgroups of order $ n. $ Show that $ G $ is commutative.
[i]Marius Tărnăuceanu[/i]
2006 IMS, 3
$G$ is a group that order of each element of it Commutator group is finite. Prove that subset of all elemets of $G$ which have finite order is a subgroup og $G$.
2007 Putnam, 5
Suppose that a finite group has exactly $ n$ elements of order $ p,$ where $ p$ is a prime. Prove that either $ n\equal{}0$ or $ p$ divides $ n\plus{}1.$
2005 VJIMC, Problem 2
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
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$).
2008 IberoAmerican Olympiad For University Students, 7
Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$).
Prove that each subgroup of $A$ of finite index is isomorphic to $A$.
1989 IMO Longlists, 82
Let $ A$ be a set of positive integers such that no positive integer greater than 1 divides all the elements of $ A.$ Prove that any sufficiently large positive integer can be written as a sum of elements of $ A.$ (Elements may occur several times in the sum.)
2006 Petru Moroșan-Trident, 1
Let be a natural number $ n\ge 4, $ and a group $ G $ for which the applications $ \iota ,\eta : G\longrightarrow G $ defined by $ \iota (g) =g^n ,\eta (g) =g^{2n} $ are endomorphisms. Prove that $ G $ is commutative if $ \iota $ is injective or surjective.
[i]Gh. Andrei[/i]
2007 Nicolae Păun, 1
Consider a finite group $ G $ and the sequence of functions $ \left( A_n \right)_{n\ge 1} :G\longrightarrow \mathcal{P} (G) $ defined as $ A_n(g) = \left\{ x\in G|x^n=g \right\} , $ where $ \mathcal{P} (G) $ is the power of $ G. $
[b]a)[/b] Prove that if $ G $ is commutative, then for any natural numbers $ n, $ either $ A_n(g) =\emptyset , $ or $ \left| A_n(g) \right| =\left| A_n(1) \right| . $
[b]b)[/b] Provide an example of what $ G $ could be in the case that there exists an element $ g_0 $ of $ G $ and a natural number $ n_0 $ such that $ \left| A_{n_0}\left( g_0 \right) \right| >\left| A_{n_0}(1) \right| . $
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
2012 Today's Calculation Of Integral, 826
Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.
1994 Hungary-Israel Binational, 4
An [i]$ n\minus{}m$ society[/i] is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every [i]$ n_0\minus{}m_0$ society[/i] contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.
2012 Putnam, 2
Let $*$ be a commutative and associative binary operation on a set $S.$ Assume that for every $x$ and $y$ in $S,$ there exists $z$ in $S$ such that $x*z=y.$ (This $z$ may depend on $x$ and $y.$) Show that if $a,b,c$ are in $S$ and $a*c=b*c,$ then $a=b.$