A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy] $\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.

The sum to infinity of the terms of an infinite geometric progression is $ 6$. The sum of the first two terms is $ 4\frac {1}{2}$. The first term of the progression is: $ \textbf{(A)}\ 3 \text{ or } 1\frac {1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2\frac {1}{2} \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 9 \text{ or } 3$

$AB$ and $CD$ are segments lying on the two sides of an angle whose vertex is $O$. $A$ is between $O$ and $B$, and $C$ is between $O$ and $D$ . The line connecting the midpoints of the segments $AD$ and $BC$ intersects $AB$ at $M$ and $CD$ at $N$. Prove that $\frac{OM}{ON}=\frac{AB}{CD}$ (V Senderov)

Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? $ \textbf{(A) } \frac 67 \qquad \textbf{(B) } \frac {13}{14} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \frac {14}{13} \qquad \textbf{(E) } \frac 76$

Given triangle $ABC$ and the points$D,E\in \left( BC \right)$, $F,G\in \left( CA \right)$, $H,I\in \left( AB \right)$ so that $BD=CE$, $CF=AG$ and $AH=BI$. Note with $M,N,P$ the midpoints of $\left[ GH \right]$, $\left[ DI \right]$ and $\left[ EF \right]$ and with ${M}'$ the intersection of the segments $AM$and $BC$. a) Prove that $\frac{B{M}'}{C{M}'}=\frac{AG}{AH}\cdot \frac{AB}{AC}$. b) Prove that the segments$AM$, $BN$ and $CP$ are concurrent.

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.

If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression $\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad$ $\textbf{(B) }\text{if and only if }a_1=a_2\qquad$ $\textbf{(C) }\text{if and only if }a_1=1\qquad$ $\textbf{(D) }\text{if and only if }a_2=1\qquad $ $\textbf{(E) }\text{if and only if }a_1=a_2=1$

Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent. [i]Bogdan Enescu, Romania[/i]

Let $ ABC $ be a triangle whose vertices have integer coordinates and inside of which there is exactly one point $ O $ with integer coordinates. Let $ D $ be the intersection of the lines $ BC $ and $ AO. $ Find the largest possible value of $ \frac {\overline{AO}} {\overline{OD}} $.

Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

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 $\textbf{(A) }12\qquad\textbf{(B) }10\qquad\textbf{(C) }5\qquad\textbf{(D) }3\qquad \textbf{(E) }2$

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.

We call a set $ A$ a good set if it has the following properties: 1. $ A$ consists circles in plane. 2. No two element of $ A$ intersect. Let $ A,B$ be two good sets. We say $ A,B$ are equivalent if we can reach from $ A$ to $ B$ by moving circles in $ A$, making them bigger or smaller in such a way that during these operations each circle does not intersect with other circles. Let $ a_{n}$ be the number of inequivalent good subsets with $ n$ elements. For example $ a_{1}\equal{} 1,a_{2}\equal{} 2,a_{3}\equal{} 4,a_{4}\equal{} 9$. [img]http://i5.tinypic.com/4r0x81v.png[/img] If there exist $ a,b$ such that $ Aa^{n}\leq a_{n}\leq Bb^{n}$, we say growth ratio of $ a_{n}$ is larger than $ a$ and is smaller than $ b$. a) Prove that growth ratio of $ a_{n}$ is larger than 2 and is smaller than 4. b) Find better bounds for upper and lower growth ratio of $ a_{n}$.

Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$. [list] a) Show that there is exactly one point $S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$. b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$. [/list]

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 $100m+10n+p.$

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.

Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by \begin{align*} f(0) &= 1\\ f(2x) &= \lfloor\phi f(x)\rfloor\\ f(2x+1) &= f(2x) + f(x). \end{align*} Find the remainder when $f(2007)$ is divided by $2008$.

Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}