Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.

In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}$. Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E$. The ratio $DE: EF$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? $ \textbf{(A)}\ 2: 1 \qquad \textbf{(B)}\ 3: 1 \qquad \textbf{(C)}\ 4: 1 \qquad \textbf{(D)}\ 16: 3 \qquad \textbf{(E)}\ 6: 1$

In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.

A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch? Note: $1$ mile= $5280$ feet [asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,1)--(5.5,1)); D((-1.5,0)--(5.5,0),dashed); D((-1.5,-1)--(5.5,-1)); [/asy] $\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$

Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$, and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$. .

Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that \[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]

In a triangle $ ABC$ with $ \angle A\equal{}90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.

$ABC$ is a non-isosceles triangle. $T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$,$T_C$ analogously). $I_A$ is the ex-center relative to the side BC (define $I_B$,$I_C$ analogously). $X_A$ is the mid-point of $I_BI_C$ (define $X_B$,$X_C$ analogously). Show that $X_AT_A$,$X_BT_B$,$X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$.

in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$. a) in what ratio does $EF$ cut the digonals?(13 points) b) find $\frac{AF}{FD}$.(5 points)

Let $ABC$ be a triangle with $AB<AC$ and let $M $ be the midpoint of $BC$. $MI$ ($I$ incenter) intersects $AB$ at $D$ and $CI$ intersects the circumcircle of $ABC$ at $E$. Prove that $\frac{ED }{ EI} = \frac{IB }{IC}$ [img]https://cdn.artofproblemsolving.com/attachments/0/5/4639d82d183247b875128a842a013ed7415fba.jpg[/img] [hide=.][url=http://artofproblemsolving.com/community/c6h602657p10667541]source[/url], translated by Antreas Hatzipolakis in fb, corrected by me in order to be compatible with it's figure[/hide]

One side of a given triangle is $ 18$ inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is: $ \textbf{(A)}\ 6\sqrt {6} \qquad \textbf{(B)}\ 9\sqrt {2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 6\sqrt {3} \qquad \textbf{(E)}\ 9$

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

In $ \triangle ABC$ we have $ AB \equal{} 7$, $ AC \equal{} 8$, and $ BC \equal{} 9$. Point $ D$ is on the circumscribed circle of the triangle so that $ \overline{AD}$ bisects $ \angle BAC$. What is the value of $ AD/CD$? $ \textbf{(A)}\ \frac{9}{8}\qquad \textbf{(B)}\ \frac{5}{3}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{17}{7}\qquad \textbf{(E)}\ \frac{5}{2}$

Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$, let $P$ be a point on $BC$, and let $Q$ be the point (other than $A$) where the line through $A$ and $P$ intersects the circumcircle of $ABC$. Prove that \[PQ\le \frac{25}{4\sqrt{6}}.\]

Let $ABC$ be a right-angled triangle ($\angle C = 90^\circ$) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$, respectively, meet at point $F$. Find the ratio $S_{ABF}:S_{ABC}$. Note: $S_{\alpha}$ means the area of $\alpha$.

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? ${ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{\sqrt{6}}{2} \qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D}}\ 2 \qquad\textbf{(E)}\ \frac{3\sqrt{2}}{2} $

Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$

Semicircles $POQ$ and $ROS$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? [asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label("$ P(-1,1) $",(-2.57,2.17),SE*lsf); label("$ Q(1,1) $",(1.55,2.21),SE*lsf); label("$ R(-1,-1) $",(-2.72,-1.45),SE*lsf); label("$S(1,-1)$",(1.59,-1.49),SE*lsf); dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] $ \textbf{(A)}\ \frac{\sqrt 2}4 \qquad\textbf{(B)}\ \frac 12 \qquad\textbf{(C)}\ \frac{2}{\pi} \qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2} $

Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.

Three real numbers $x$, $y$, and $z$ are such that $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$. Determine the value of $x/y$.

A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.500$. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $.503$. What's the largest number of matches she could've won before the weekend began?

Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively. a) Prove that $BE=EZ=ZC$. b) Find the ratio of the areas of the triangles $BDE$ to $ABC$