Points $E$ and $F$ are located on square $ABCD$ so that $\Delta BEF$ is equilateral. What is the ratio of the area of $\Delta DEF$ to that of $\Delta ABE$? [asy] pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=B+2*dir(165), E=intersectionpoint(B--X, A--D), Y=B+2*dir(105), F=intersectionpoint(B--Y, D--C); draw(B--C--D--A--B--F--E--B); pair point=(0.5,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy] $\textbf{(A)}\; \frac43\qquad \textbf{(B)}\; \frac32\qquad \textbf{(C)}\; \sqrt3\qquad \textbf{(D)}\; 2\qquad \textbf{(E)}\; 1+\sqrt3\qquad$

Trapezoid $ ABCD$ has bases $ \overline{AB}$ and $ \overline{CD}$ and diagonals intersecting at $ K$. Suppose that $ AB\equal{}9$, $ DC\equal{}12$, and the area of $ \triangle AKD$ is $ 24$. What is the area of trapezoid $ ABCD$? $ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 94 \qquad \textbf{(C)}\ 96 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 100$

A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces. [i](6 points)[/i]

Two cyclists, $ k$ miles apart, and starting at the same time, would be together in $ r$ hours if they traveled in the same direction, but would pass each other in $ t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: $ \textbf{(A)}\ \frac {r \plus{} t}{r \minus{} t} \qquad\textbf{(B)}\ \frac {r}{r \minus{} t} \qquad\textbf{(C)}\ \frac {r \plus{} t}{r} \qquad\textbf{(D)}\ \frac {r}{t} \qquad\textbf{(E)}\ \frac {r \plus{} k}{t \minus{} k}$

$AB$ 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)

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio $y:x$ is: $\textbf{(A) }23:17\qquad \textbf{(B) }17y:23\qquad \textbf{(C) }23x:17\qquad$ $\textbf{(D) }\text{dependent upon the cost}\qquad \textbf{(E) }\text{none of these.}$

Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

Consider the square $ABCD$ in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square $ABCD.$

Let $a$ and $b$ be distinct real numbers for which \[\dfrac ab+\dfrac{a+10b}{b+10a}=2.\] Find $\dfrac ab$. $\textbf{(A) }0.6\qquad\textbf{(B) }0.7\qquad\textbf{(C) }0.8\qquad\textbf{(D) }0.9\qquad\textbf{(E) }1$

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class? $\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$

Define a sequence of real numbers $ a_1$, $ a_2$, $ a_3$, $ \dots$ by $ a_1 = 1$ and $ a_{n + 1}^3 = 99a_n^3$ for all $ n \ge 1$. Then $ a_{100}$ equals $ \textbf{(A)}\ 33^{33} \qquad \textbf{(B)}\ 33^{99} \qquad \textbf{(C)}\ 99^{33} \qquad \textbf{(D)}\ 99^{99} \qquad \textbf{(E)}\ \text{none of these}$

Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.