Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.

Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$? $\textbf{(A) }-2\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6\qquad$

The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.

Let $F=.48181\cdots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }29\qquad\textbf{(D) }57\qquad \textbf{(E) }126$

Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$. Determine the maximum possible value of degree of $T(x)$

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

Let $c_i$ denote the $i$th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute \[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\] (Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$)

Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$. (1) Find the probability such that $s_n$ is divisible by 4. (2) Find the probability such that $s_n$ is divisible by 6. (3) Find the probability such that $s_n$ is divisible by 7. Last Edited Thanks, jmerry & JBL

By adding the same constant to $20,50,100$ a geometric progression results. The common ratio is: $ \textbf{(A)}\ \frac53 \qquad\textbf{(B)}\ \frac43\qquad\textbf{(C)}\ \frac32\qquad\textbf{(D)}\ \frac12\qquad\textbf{(E)}\ \frac13 $

Prove that if $|x| < 1$, then \[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]

Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$ be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.

The sum of an infinite geometric series with common ratio $r$ such that $|r|<1$, is $15$, and the sum of the squares of the terms of this series is $45$. The first term of the series is $\textbf{(A) }12\qquad\textbf{(B) }10\qquad\textbf{(C) }5\qquad\textbf{(D) }3\qquad \textbf{(E) }2$

What is the remainder when $ 3^0\plus{}3^1\plus{}3^2\plus{}\ldots\plus{}3^{2009}$ is divided by $ 8$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 6$

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

If $X=1+x+x^2+x^3+\cdots$ and $Y=1+y+y^2+y^3+\cdots$, what is $1+xy+x^2y^2+x^3y^3+\cdots$ in terms of $X$ and $Y$ only?

Echo the gecko starts on the point $(0, 0)$ in the 2D coordinate plane. Every minute, starting at the end of the first minute, he'll teleport $1$ unit up, left, right, or down with equal probability. Echo dies the moment he lands on a point that is more than $1$ unit away from the origin. The average number of minutes he'll live can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [i]Team #2[/i]

In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a [i]Marble distribution[/i] according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student. Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.

Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.

Let G (n) = max | A(n) |, where A(n) ranges over all subsets of {1,2,...,n} and contains no three-member geometric series, ie, there is no $x, y, z \in A$ such that x < y < z and xz = y^2. Prove that $\lim_{n \to \infty} \frac{G (n)}{n}$ exists.

If $a+\log_2 3,a+\log_4 3,a+\log_8 3$ are a geometric series, then the common ratio is________.

For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.