[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. [b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.

Find the value of $\frac{2014^3-2013^3-1}{2013\times 2014}$. $ \textbf{(A) }3\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }11 $

In a convex quadrilateral $ABCD$ whose diagonals intersect at point $E$, the equalities$$\dfrac{|AB|}{|CD|}=\dfrac{|BC|}{|AD|}=\sqrt{\dfrac{|BE|}{|ED|}}$$hold. Prove that $ABCD$ is either a paralellogram or a cyclic quadrilateral

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers $a_{-1},\ldots,a_{-k}$ on a blackboard, so he decided to do the following process just for fun: if there are real numbers $a_{-k},\ldots,a_{n-1}$ on the blackboard, then he computes the polynomial $$P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).$$ He then writes a real number $a_n$, where $$a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.$$ If $a_n$ is undefined (that is, $P_n(i)+P_n(-i)=0$), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists $m>n\ge0$ such that $am=an$), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)

Let $\omega$ be the incircle of $\triangle ABC$ and $D,E,F$ be the tangency points on $BC ,CA, AB$. In $\triangle DEF$ let the altitudes from $E,F$ to $FD,DE$ intersect $AB, AC$ at $X ,Y$. Prove that the second intersection of $(AEX)$ and $(AFY)$ lies on $\omega$

[b]a)[/b] $ \lim_{n\to\infty } \sum_{j=1}^n\frac{n}{n^2+n+j} =1 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n- \sum_{j=1}^n\frac{n^2}{n^2+n+j} \right) =3/2 $ [i]Cristinel Mortici[/i]

A circle $\omega$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $XP$ and the circle $XOP$ meet $\omega$ for a second time at points $X_1$, $X_2$ respectively. Prove that all lines $X_1X_2$ are parallel.

A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\angle ACB$ if $\angle ABC = 40^o$.

Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.

Integers $ a,$ $ b,$ $ c,$ and $ d,$ not necessarily distinct, are chosen independantly and at random from $ 0$ to $ 2007,$ inclusive. What is the probability that $ ad \minus{} bc$ is even? $ \textbf{(A)}\ \frac {3}{8}\qquad \textbf{(B)}\ \frac {7}{16}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {9}{16}\qquad \textbf{(E)}\ \frac {5}{8}$

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and \[2^ap^b=(p+2)^c-1.\]

Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, then the area of the original triangle is $\textbf{(A) }180\qquad\textbf{(B) }190\qquad\textbf{(C) }200\qquad\textbf{(D) }210\qquad \textbf{(E) }240$

Let $F_0 = 0, F_{1} = 1$, and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$. Find $A + B$.

Let $n$ be the answer to this problem. Hexagon $ABCDEF$ is inscribed in a circle of radius $90$. The area of $ABCDEF$ is $8n$, $AB = BC = DE = EF$, and $CD = FA$. Find the area of triangle $ABC$:

Let $d(n)$ be number of prime divisors of $n$. Prove that one can find $k,m$ positive integers for any positive integer $n$ such that $k-m=n$ and $d(k)-d(m)=1$

In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

The product of the positive real numbers $x, y, z$ is 1. Show that if \[ \frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z \]then \[ \frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k} \] for all positive integers $k$.

Given a regular pentagon, its five diagonals are drawn. How many triangles do appear in the figure? Classify the set of triangles in classes of equal triangles.

Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

Define a sequence $F_n$ such that $F_1 = 1$, $F_2 = x$, $F_{n+1} = xF_n + yF_{n-1}$ where and $x$ and $y$ are positive integers. Suppose $\frac{1}{F_k}= \sum_{n=1}^{\infty}\frac{F_n}{d^n}$ has exactly two solutions $(d, k)$ with $d > 0$ is a positive integer. Find the least possible positive value of $d$.

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

Let $n$ be a fixed positive integer. Ben is playing a computer game. The computer picks a tree $T$ such that no vertex of $T$ has degree $2$ and such that $T$ has exactly $n$ leaves, labeled $v_1,\ldots, v_n$. The computer then puts an integer weight on each edge of $T$, and shows Ben neither the tree $T$ nor the weights. Ben can ask queries by specifying two integers $1\leq i < j \leq n$, and the computer will return the sum of the weights on the path from $v_i$ to $v_j$. At any point, Ben can guess whether the tree's weights are all zero. He wins the game if he is correct, and loses if he is incorrect. (a) Show that if Ben asks all $\binom n2$ possible queries, then he can guarantee victory. (b) Does Ben have a strategy to guarantee victory in less than $\binom n2$ queries? [i]Brandon Wang[/i]