Found problems: 216
Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.
In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.
[i]Proposed by Lewis Chen[/i]
Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.
Cyclic pentagon $ ABCDE$ has a right angle $ \angle{ABC} \equal{} 90^{\circ}$ and side lengths $ AB \equal{} 15$ and $ BC \equal{} 20$. Supposing that $ AB \equal{} DE \equal{} EA$, find $ CD$.
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
[i]Ray Li[/i]
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?
Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}$. What is $ r/s$?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=3;
pair B=(0,0), C=(3,0), D=(3,3), A=(0,3);
pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6);
pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0];
pair[] dots={A,B,C,D,Ep,F};
draw(A--F);
draw(Circle(Ep,5/3));
draw(A--B--C--D--cycle);
dot(dots);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",Ep,E);
label("$F$",F,NW);[/asy]$ \textbf{(A) } \frac {1}{2}\qquad \textbf{(B) } \frac {5}{9}\qquad \textbf{(C) } \frac {3}{5}\qquad \textbf{(D) } \frac {5}{3}\qquad \textbf{(E) } \frac {9}{5}$
A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.
[asy]
size(150);
defaultpen(linewidth(0.8));
draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1));
draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4"));
draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4"));
label("$5$",(0,5/2),W);
label("$8$",(4,0),S);
[/asy]
In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$.
A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$.
Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.
Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then the area of the smaller circle is
[asy]
size(250);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, Q=(0,-3sqrt(2)), P=(0,-6sqrt(2)), A=(-4/3,3.77-6sqrt(2)), B=(-8/3,7.54-6sqrt(2)), C=(4/3,3.77-6sqrt(2)), D=(8/3,7.54-6sqrt(2));
draw(Arc(O,2sqrt(2),0,360));
draw(Arc(Q,sqrt(2),0,360));
dot(A);
dot(B);
dot(C);
dot(D);
dot(P);
draw(B--A--P--C--D);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$A'$",C,dir(C));
label("$B'$",D,dir(D));
label("$P$",P,S);[/asy]
$ \textbf{(A)}\ 1.44\pi\qquad\textbf{(B)}\ 2\pi\qquad\textbf{(C)}\ 2.56\pi\qquad\textbf{(D)}\ \sqrt{8}\pi\qquad\textbf{(E)}\ 4\pi $
Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is
$\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$
Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$?
[asy]unitsize(4cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1);
pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0];
pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
draw(A--B--C--D--cycle);
draw(D--F);
draw(C--E);
draw(B--H);
draw(A--G);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",E,NNW);
label("$F$",F,ENE);
label("$G$",G,SSE);
label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.
A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(1));
draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4));
[/asy]
In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$.
Find the distance between centers of circles $\omega_1$ and $\omega_2$.
[i]I. Voronovich[/i]
Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.
In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \overline{AB}$ and $ \overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \overline{BD}$, $ \overline{BE}$, and the minor arc connecting $ D$ and $ E$?
[asy]
defaultpen(linewidth(0.8));
pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B;
fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray);
clip(B--Arc(O, 2, 30, 60)--cycle);
draw(Circle(origin, 2));
draw((-2,0)--(2,0)^^(0,-2)--(0,2));
draw(A--D^^C--E);
label("$A$", A, dir(point--A));
label("$C$", C, dir(point--C));
label("$O$", O, dir(point--O));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$B$", B, SW);[/asy]
$ \textbf{(A) } \frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad \textbf{(B) } \frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad \textbf{(C) } \pi\left(2 \minus{} \sqrt {3}\right) \qquad \textbf{(D) } \frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\
\qquad \indent \textbf{(E) } \frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}$
Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals:
$\textbf{(A)}\ \dfrac{3a}{5} \qquad
\textbf{(B)}\ \dfrac{5a}{8} \qquad
\textbf{(C)}\ \dfrac{3a}{8} \qquad
\textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad
\textbf{(E)}\ \dfrac{a}{2}$
Two circles with radii 1 and 2 touch each other and a line as in the figure. In the region between the circles and the line, there is a circle with radius $ r$ which touches the two circles and the line. What is $ r$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage2.jpg[/img]
A. 1/3
B. $ \frac {1}{\sqrt {5}}$
C. $ \sqrt {3} \minus{} \sqrt {2}$
D. $ 6 \minus{} 4 \sqrt {2}$
E. None of these