Found problems: 216
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.
The altitude drawn to the base of an isosceles triangle is $ 8$, and the perimeter $ 32$. The area of the triangle is:
$ \textbf{(A)}\ 56\qquad
\textbf{(B)}\ 48\qquad
\textbf{(C)}\ 40\qquad
\textbf{(D)}\ 32\qquad
\textbf{(E)}\ 24$
In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is
[asy]
size(120); defaultpen(linewidth(0.7));
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);
draw(Circle(O, 5));
draw(A--B^^C--D--F);
dot(O^^A^^B^^C^^D^^E^^F);
markscalefactor=0.05;
draw(rightanglemark(B, O, D));
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$F$", F, dir(O--F));
label("$O$", O, NW);
label("$E$", E, SE);[/asy]
$\textbf{(A)}\ 23\pi \qquad
\textbf{(B)}\ \dfrac{47}{2}\pi \qquad
\textbf{(C)}\ 24\pi \qquad
\textbf{(D)}\ \dfrac{49}{2}\pi \qquad
\textbf{(E)}\ 25\pi$
Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3,4,12, and 13, respectively, and $\measuredangle CBA$ is a right angle. The area of the quadrilateral is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real r=degrees((12,5)), s=degrees((3,4));
pair D=origin, A=(13,0), C=D+12*dir(r), B=A+3*dir(180-(90-r+s));
draw(A--B--C--D--cycle);
markscalefactor=0.05;
draw(rightanglemark(A,B,C));
pair point=incenter(A,C,D);
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("$3$", A--B, dir(A--B)*dir(-90));
label("$4$", B--C, dir(B--C)*dir(-90));
label("$12$", C--D, dir(C--D)*dir(-90));
label("$13$", D--A, dir(D--A)*dir(-90));[/asy]
$\text{(A)} \ 32 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 39 \qquad \text{(D)} \ 42 \qquad \text{(E)} \ 48$
Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$ meters, and $\angle BAC=45^{\circ}$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers?
$ \textbf{(A)}\ 30\text{ and }31\qquad\textbf{(B)}\ 31\text{ and }32\qquad\textbf{(C)}\ 32\text{ and }33\qquad\textbf{(D)}\ 33\text{ and }34\qquad\textbf{(E)}\ 34\text{ and }35$
A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly's path if both insects take the shortest path possible?
Given the areas of the three squares in the figure, what is the area of the interior triangle?
[asy]
real r=22.61986495;
pair A=origin, B=(12,0), C=(12,5);
draw(A--B--C--cycle);
markscalefactor=0.1;
draw(rightanglemark(C, B, A));
draw(scale(12)*shift(0,-1)*unitsquare);
draw(scale(5)*shift(12/5,0)*unitsquare);
draw(scale(13)*rotate(r)*unitsquare);
pair P=shift(0,-1)*(13/sqrt(2) * dir(r+45)), Q=(14.5,1.2), R=(6, -7);
label("169", P, N);
label("25", Q, N);
label("144", R, N);
[/asy]
$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 1800$
Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?
In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.
In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
[asy]
unitsize(150);
pair A,B,C,D,E,F,G,H,J,K;
A=(1,0); B=(0,0); C=(0,1); D=(1,1);
draw(A--B--C--D--A);
E=(2-sqrt(3),0); F=(2-sqrt(3),1);
draw(E--F);
G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1);
K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0);
draw(G--H--K--J--G);
label("$A$",A,SE);
label("$B$",B,SW);
label("$C$",C,NW);
label("$D$",D,NE);
label("$E$",E,S);
label("$F$",F,N);
label("$G$",G,E);
label("$H$",H,N);
label("$K$",K,W);
label("$J$",J,S);
[/asy]
$ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?
$ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $
Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$
Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1\]
\[\sqrt{z-b}+\sqrt{x-b}=1\]
\[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x,y,z)$ in real numbers.
It was proposed by Poland. Have fun! :lol:
Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$
While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at $ 10 \text{m/s}$, and you manage a three-second head start over the raptor. If the raptor runs at $ 15\sqrt{2} \text{m/s}$, but only runs either north or east at any given time, how many seconds do you have until it devours you?
In $ \triangle ABC$ , $ AB \equal{} 13$, $ AC \equal{} 5$, and $ BC \equal{} 12$. Points $ M$ and $ N$ lie on $ \overline{AC}$ and $ \overline{BC}$, respectively, with $ CM \equal{} CN \equal{} 4$. Points $ J$ and $ K$ are on $ \overline{AB}$ so that $ \overline{MJ}$ and $ \overline{NK}$ are perpendicular to $ \overline{AB}$. What is the area of pentagon $ CMJKN$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair C=(0,0), B=(12,0), A=(0,5), M=(0,4), Np=(4,0);
pair K=foot(Np,A,B), J=foot(M,A,B);
draw(A--B--C--cycle);
draw(M--J);
draw(Np--K);
label("$C$",C,SW);
label("$A$",A,NW);
label("$B$",B,SE);
label("$N$",Np,S);
label("$M$",M,W);
label("$J$",J,NE);
label("$K$",K,NE);[/asy]$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ \frac{81}{5} \qquad
\textbf{(C)}\ \frac{205}{12} \qquad
\textbf{(D)}\ \frac{240}{13} \qquad
\textbf{(E)}\ 20$
In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 2 \sqrt {3}\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 4 \sqrt {2}$
Each side of triangle $ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, in the same unit, is:
${{ \textbf{(A)}\ \sqrt{18} \qquad\textbf{(B)}\ \sqrt{28} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \sqrt{63} }\qquad\textbf{(E)}\ \sqrt{98} } $
The base of isosceles $\triangle{ABC}$ is $24$ and its area is $60$. What is the length of one of the congruent sides?
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 13 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 18$
Convex quadrilateral $ABCD$ has $AB = 3, BC = 4, CD = 13, AD = 12,$ and $\angle ABC = 90^\circ,$ as shown. What is the area of the quadrilateral?
[asy]
unitsize(.4cm);
defaultpen(linewidth(.8pt)+fontsize(14pt));
dotfactor=2;
pair A,B,C,D;
C = (0,0);
B = (0,4);
A = (3,4);
D = (12.8,-2.8);
draw(C--B--A--D--cycle);
draw(rightanglemark(C,B,A,20));
dot("$A$",A,N);
dot("$B$",B,NW);
dot("$C$",C,SW);
dot("$D$",D,E);
[/asy]
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 58.5 $
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?
A triangle with sides of $ 5$, $ 12$, and $ 13$ has both an inscibed and a circumscribed circle. What is the distance between the centers of those circles?
$ \textbf{(A)}\ \frac{3\sqrt{5}}{2}\qquad
\textbf{(B)}\ \frac{7}{2}\qquad
\textbf{(C)}\ \sqrt{15}\qquad
\textbf{(D)}\ \frac{\sqrt{65}}{2}\qquad
\textbf{(E)}\ \frac{9}{2}$
The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse?
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$
The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball.
An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus?
$ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$
[asy]unitsize(1.4cm);
defaultpen(linewidth(.8pt));
dotfactor=3;
real r1=1.0, r2=1.8;
pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90);
pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0];
pair[] points={X,O,Y,Z};
filldraw(Circle(O,r2),mediumgray,black);
filldraw(Circle(O,r1),white,black);
dot(points);
draw(X--Y--O--cycle--Z);
label("$O$",O,SSW,fontsize(10pt));
label("$Z$",Z,SW,fontsize(10pt));
label("$Y$",Y,N,fontsize(10pt));
label("$X$",X,NE,fontsize(10pt));
defaultpen(fontsize(8pt));
label("$c$",midpoint(O--Z),W);
label("$d$",midpoint(Z--Y),W);
label("$e$",midpoint(X--Y),NE);
label("$a$",midpoint(X--Z),N);
label("$b$",midpoint(O--X),SE);[/asy]
Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.