Let $(A,+,\cdot)$ be a ring with unity, having the following property: for all $x\in A$ either $x^2=1$ or $x^n=0$ for some $n\in\mathbb N$. Show that $A$ is a commutative ring.

Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$.

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]

The remainder of $1991^{2000}$ module $10^6$ is________.

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)

Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]

Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$ [i]Marian Andronache[/i]

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.

Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the [i]weight of a vertex [/i]the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero. [i]Proposed by W. Tutte[/i]

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$

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

Let $ m $ be a non-negative ineger, $ n\ge 2 $ be a natural number, $ A $ be a ring which has exactly $ n $ elements, and an element $ a $ of $ A $ such that $ 1-a^k $ is invertible, for all $ k\in\{ m+1,m+2,...,m+n-1\} . $ Prove that $ a $ is nilpotent.

Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point. [i]Vasile Pop[/i]

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are: (1) A player cannot choose an element that has been chosen by either player on any previous turn. (2) A player can only choose an element that commutes with all previously chosen elements. (3) A player who cannot choose an element on his/her turn loses the game. Patniss takes the first turn. Which player has a winning strategy?

Let $\gamma,\gamma_0,\gamma_1,\gamma_2$ be four circles in plane,such that $\gamma_i$ is interiorly tangent to $\gamma$ in point $A_i$,and $\gamma_i$ and $\gamma_{i+1}$ are exteriorly tangent in point $B_{i+2}$,$i=0,1,2$(the indexes are reduced modulo $3$).The tangent in $B_i$,common for circles $\gamma_{i-1}$ and $\gamma_{i+1}$,intersects circle $\gamma$ in point $C_i$,situated in the opposite semiplane of $A_i$ with respect to line $A_{i-1}A_{i+1}$.Prove that the three lines $A_iC_i$ are concurrent.

$G$ be a group and $H\le G$. Then which of the following are true? $\textbf{(A)}~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H$ $\textbf{(B)}~\exists G,H\text{ and }H\le G\text{ with }H\cong G$ $\textbf{(C)}~\text{All subgroups are normal, then }G\text{ is abelian.}$ $\textbf{(D)}~\text{None of the above}$

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]

Let be the set $ G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} $ and a function $ \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} $ having the property that the operation $ *:G^2\longrightarrow G $ defined as $$ (a,b)*(c,d)=(ac,bc+d\varphi (a)) $$ is associative. [b]a)[/b] Show that $ (G,*) $ is a group. [b]b)[/b] Describe $ \varphi , $ knowing that $(G,*) $ is a commutative group. [i]Marius Perianu[/i]

a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$ b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality? c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.

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

[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]

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

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 $A$.