In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?

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$

In the triangle $ ABC $ on the sides $ BC $, $ CA $, $ AB $, the points $ D $, $ E $, $ F $ are chosen respectively in such a way that $$ BD \colon DC = CE \colon EA = AF \colon FB = k,$$ where $k$ is a given positive number. Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle $ DEF $

In triangle $ABC$, angle $B$ is different from a right angle, $AB : BC = k$. Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ wrt $AB$ and $BC$ intersect line $AC$ at points $D$ and $E$. Find $BD : BE$.

The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary? $ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$

Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of triangle $DFE$ to the area of quadrilateral $ABEF$? $ \textbf{(A)}\ 1:2 \qquad\textbf{(B)}\ 1:3 \qquad\textbf{(C)}\ 1:5 \qquad\textbf{(D)}\ 1:6 \qquad\textbf{(E)}\ 1:7 $

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

Triangle $ ABC$ is a right triangle with $ \angle ACB$ as its right angle, $ m\angle ABC \equal{} 60^\circ$, and $ AB \equal{} 10$. Let $ P$ be randomly chosen inside $ \triangle ABC$, and extend $ \overline{BP}$ to meet $ \overline{AC}$ at $ D$. What is the probability that $ BD > 5\sqrt2$? [asy]import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P); draw(A--C--B--cycle); draw(B--D); dot(P); label("A",A,S); label("D",D,S); label("C",C,S); label("P",P,NE); label("B",B,N);[/asy] $ \textbf{(A)}\ \frac {2 \minus{} \sqrt2}{2} \qquad \textbf{(B)}\ \frac {1}{3} \qquad \textbf{(C)}\ \frac {3 \minus{} \sqrt3}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {5 \minus{} \sqrt5}{5}$

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)

$ABC$ is a triangle $E, F$ are points in $AB$, such that $AE = EF = FB$ $D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$ $AD$ is perpendicular to $CF$. The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$) Determine the ratio $\frac{DB}{DC}$. %Edited!%

In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.

If $ 1 \minus{} y$ is used as an approximation to the value of $ \frac {1}{1 \plus{} y}$, $ |y| < 1$, the ratio of the error made to the correct value is: $ \textbf{(A)}\ y \qquad \textbf{(B)}\ y^2 \qquad \textbf{(C)}\ \frac {1}{1 \plus{} y} \qquad \textbf{(D)}\ \frac {y}{1 \plus{} y} \qquad \textbf{(E)}\ \frac {y^2}{1 \plus{} y}\qquad$

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? $\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4$

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

Let $ A_1,B_1,C_1 $ be the feet of the heights of an acute triangle $ ABC. $ On the segments $ B_1C_1,C_1A_1,A_1B_1, $ take the points $ X,Y, $ respectively, $ Z, $ such that $$ \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\ \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . $$ Show that $ AX,BY,CZ, $ are concurrent.

The sides of a right triangle are $a, a+d,$ and $a+2d$, with $a$ and $d$ both positive. The ratio of $a$ to $d$ is: $ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:4 \qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 3:1\qquad\textbf{(E)}\ 3:4 $

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

Country $\mathcal{A}$ has $c\%$ of the world's population and owns $d\%$ of the world's wealth. Country $\mathcal{B}$ has $e\%$ of the world's population and $f\%$ of its wealth. Assume that the citizens of $\mathcal{A}$ share the wealth of $\mathcal{A}$ equally, and assume that those of $\mathcal{B}$ share the wealth of $\mathcal{B}$ equally. Find the ratio of the wealth of a citizen of $\mathcal{A}$ to the wealth of a citizen of $\mathcal{B}$. $ \textbf{(A)}\ \frac{cd}{ef} \qquad\textbf{(B)}\ \frac{ce}{df} \qquad\textbf{(C)}\ \frac{cf}{de} \qquad\textbf{(D)}\ \frac{de}{cf} \qquad\textbf{(E)}\ \frac{df}{ce} $

Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

Let $ ABC$ be a triangle. The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$. The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$. The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$. The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$. Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram. [i]Remark.[/i] The $ A$[i]-excircle[/i] of a triangle $ ABC$ is defined as the circle which touches the segment $ BC$ and the extensions of the segments $ CA$ and $ AB$ beyound the points $ C$ and $ B$, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle $ CAB$ and the exterior angle bisectors of the angles $ ABC$ and $ BCA$. Similarly, the $ B$-excircle and the $ C$-excircle of triangle $ ABC$ are defined. [hide="Source of the problem"][i]Source of the problem:[/i] Theorem (88) in: John Sturgeon Mackay, [i]The Triangle and its Six Scribed Circles[/i], Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.[/hide]

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21)); pair D=foot(B,A,C); pair[] ps={B,C,A,D}; draw(A--B--C--cycle); draw(B--D); draw(rightanglemark(B,D,C)); dot(ps); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); label("$3$",midpoint(A--D),NE); label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 7\sqrt3 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 14\sqrt3 \qquad \textbf{(E)}\ 42$

In a triangle $ABC$, points $D$ and $E$ are on segments $BC$ and $AC$ such that $BD=3DC$ and $AE=4EC$. Point $P$ is on line $ED$ such that $D$ is the midpoint of segment $EP$. Lines $AP$ and $BC$ intersect at point $S$. Find the ratio $BS/SD$.

Let $O$ be the center of a circle inscribed in a convex quadrilateral $ABCD$ and $|AB|= a$, $|CD|=$c. Prove that $$\frac{a}{c}=\frac{AO\cdot BO}{CO\cdot DO}.$$