A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\circ$ counterclockwise and travels $\frac 12$ a unit to $\left(1, \frac 12 \right)$. If it continues in this fashion, each time making a $90^\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest? $\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)$

In a geometric series of positive terms the difference between the fifth and fourth terms is $576$, and the difference between the second and first terms is $9$. What is the sum of the first five terms of this series? $ \textbf{(A)}\ 1061 \qquad\textbf{(B)}\ 1023 \qquad\textbf{(C)}\ 1024 \qquad\textbf{(D)}\ 768 \qquad\textbf{(E)}\ \text{none of these} $

The pattern in the figure below continues inward infinitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area. [asy] defaultpen(linewidth(0.8)); pen blu = rgb(0,112,191); real r=sqrt(3); fill((8,0)--(0,8r)--(-8,0)--cycle, blu); fill(origin--(4,4r)--(-4,4r)--cycle, white); fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu); fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]

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 $S?$ $\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that \[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \] Give an example of two such progressions having at least five terms. [i]Mihai Baluna[/i]

The geometric series $ a \plus{} ar \plus{} ar^{2} \plus{} ...$ has a sum of $ 7$, and the terms involving odd powers of $ r$ have a sum of $ 3$. What is $ a \plus{} r$? $ \textbf{(A)}\ \frac {4}{3}\qquad \textbf{(B)}\ \frac {12}{7}\qquad \textbf{(C)}\ \frac {3}{2}\qquad \textbf{(D)}\ \frac {7}{3}\qquad \textbf{(E)}\ \frac {5}{2}$

Let $ f(r) \equal{} \sum_{j \equal{} 2}^{2008} \frac {1}{j^r} \equal{} \frac {1}{2^r} \plus{} \frac {1}{3^r} \plus{} \dots \plus{} \frac {1}{2008^r}$. Find $ \sum_{k \equal{} 2}^{\infty} f(k)$.

Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If \[ S = \sum_{i = 1}^\infty r_i, \] then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$. [i]Proposed by Aaron Lin[/i]

Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.

Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and \[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\] Determine $ f \left( \frac{1}{7} \right).$

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

In $\triangle ABC$, if $A,B,C$ are geometric series, and $b^2-a^2=ac$, then $B=$________.

Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.

Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.

Square ABCD has side length 7. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. SImilarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares. [asy] size(180); pair[] A={(-1,-1),(-1,1),(1,1),(1,-1),(-1,-1)}; string[] X={"A","B","C","D"}; for(int k=0;k<10;++k) { for(int m=0;m<4;++m) { if(k==0) label("$"+X[m]+"$",A[m],A[m]); if(k==1) label("$"+X[m]+"_1$",A[m],A[m]); draw(A[m]--A[m+1]); A[m]+=3/7*(A[m+1]-A[m]); } A[4]=A[0]; }[/asy]

Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?

Evaluate $\displaystyle\sum_{n=0}^\infty \dfrac{\cos n\theta}{2^n}$, where $\cos\theta = \dfrac{1}{5}$.

Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]

The infinite geometric series of positive reals $a_1, a_2, \dots$ satisfies \[ 1 = \sum_{n=1}^\infty a_n = -\frac{1}{2013} + \sum_{n=1}^{\infty} \text{GM}(a_1, a_2, \dots, a_n) = \frac{1}{N} + a_1 \] where $\text{GM}(x_1, x_2, \dots, x_k) = \sqrt[k]{x_1x_2\cdots x_k}$ denotes the geometric mean. Compute $N$. [i]Proposed by Aaron Lin[/i]

Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum_{k=1}^n \frac{1}{a_k} < 90$.

Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$