A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer. [i]($3$ points)[/i]

A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term?

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is 3, then the sum of the first two terms of the series is $ \textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 8/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 9/2$

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

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

by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.

Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]

Cast a dice $n$ times. Denote by $X_1,\ X_2,\ \cdots ,\ X_n$ the numbers shown on each dice. Define $Y_1,\ Y_2,\ \cdots,\ Y_n$ by \[Y_1=X_1,\ Y_k=X_k+\frac{1}{Y_{k-1}}\ (k=2,\ \cdots,\ n)\] Find the probability $p_n$ such that $\frac{1+\sqrt{3}}{2}\leq Y_n\leq 1+\sqrt{3}.$ 35 points

How many of the numbers \[ a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6 \] are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$? $ \textbf{(A)}\ 121 \qquad\textbf{(B)}\ 224 \qquad\textbf{(C)}\ 275 \qquad\textbf{(D)}\ 364 \qquad\textbf{(E)}\ 375 $

Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$

The sequence $(x_n)$ is given by $x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$ Prove that for all natural numbers $n \geq 1$ the following inequality holds $x_1+x_2+...+x_n < 1$.

For geometric series $(a_n)$ with all items real, if $S_{10}=10,S_{30}=70$, then $S_{40}=$ $\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400$ Note: $S_n=\sum_{i=1}^{n}a_i$.

Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that $1$, $2$, $3$, $4$, $5$, and $6$ dots are on separate faces. In a fit of mischief, James places his $21$ dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct configuration. Regardless of the configuration of dots, both dice are unweighted and have equal probability of showing each face after being rolled. Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability.

Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)

For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.

Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8,$ and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p},$ where $m,$ $n,$ and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p.$

Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if \[ \frac{n}{810}=0.d25d25d25\ldots \]

Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.

Jordan has an infinite geometric series of positive reals whose sum is equal to $2\sqrt2 + 2$. It turns out that if Jordan squares each term of his geometric series and adds up the resulting numbers, he get a sum equal to $4$. If Jordan decides to take the fourth power of each term of his original geometric series and add up the resulting numbers, what sum will he get?

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$ $\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.

The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$.

Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]