Found problems: 339
2011 Romania National Olympiad, 2
[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]
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 $G$.
2013 Online Math Open Problems, 47
Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers
such that
\[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \]
for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$.
[i]David Yang[/i]
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.
2018 AIME Problems, 2
Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.
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.$
2010 Iran MO (3rd Round), 3
If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points)
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 Macedonia National Olympiad, 3
Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.
1985 Traian Lălescu, 1.4
Let $ A $ be a ring in which $ 1\neq 0. $ If $ a,b\in A, $ then the following affirmations are equivalent:
$ \text{(i)}\quad aba=a\wedge ba^2b=1 $
$ \text{(ii)}\quad ab=ba=1 $
$ \text{(iii)}\quad \exists !b\in A\quad aba=a $
2004 Alexandru Myller, 3
Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.
2021 CCA Math Bonanza, L3.3
Compute the smallest positive integer that gives a remainder of $1$ when divided by $11$, a remainder of $2$ when divided by $21$, and a remainder of $5$ when divided by $51$.
[i]2021 CCA Math Bonanza Lightning Round #3.3[/i]
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]
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 $a=b.$
2007 Today's Calculation Of Integral, 200
Evaluate the following definite integral.
\[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]
1996 Miklós Schweitzer, 4
Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.
2025 AIME, 15
Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.
1993 Hungary-Israel Binational, 1
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.
Suppose $k \geq 2$ is an integer such that for all $x, y \in G$ and $i \in \{k-1, k, k+1\}$ the relation $(xy)^{i}= x^{i}y^{i}$ holds. Show that $G$ is Abelian.
2019 LIMIT Category C, Problem 5
Let $G=(S^1,\cdot)$ be a group. Then its nontrivial subgroups
$\textbf{(A)}~\text{are necessarily finite}$
$\textbf{(B)}~\text{can be infinite}$
$\textbf{(C)}~\text{can be dense in }S^1$
$\textbf{(D)}~\text{None of the above}$
2011 Pre-Preparation Course Examination, 2
prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure.
[b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common.
we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.