Found problems: 216
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
$ \textbf{(A)}\ 4\plus{}\sqrt{185} \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ \sqrt{32}\plus{}\sqrt{137}$
Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$.
[i]Proposed by Evan Chen[/i]
The triangle $ABC$ has a right angle at $C.$ The point $P$ is located on segment $AC$ such that triangles $PBA$ and $PBC$ have congruent inscribed circles. Express the length $x = PC$ in terms of $a = BC, b = CA$ and $c = AB.$
Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$?
[asy]unitsize(2.5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
pair M=midpoint(A--Ep);
pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1];
pair D=B+8*dir(180+degrees(C));
dot(A);
dot(C);
dot(B);
dot(D);
draw(C--D);
draw(A--B);
draw(Circle(A,3));
draw(Circle(B,8));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,SE);
label("$E$",Ep,SSE);
label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$
There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.
In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.
A ball is placed on a plane and a point on the ball is marked.
Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling.
Prove that it is possible.
Time allowed for this problem was 90 minutes.
Three wooden equilateral triangles of side length $18$ inches are placed on axles as shown in the diagram to the right. Each axle is $30$ inches from the other two axles. A $144$-inch leather band is wrapped around the wooden triangles, and a dot at the top corner is painted as shown. The three triangles are then rotated at the same speed and the band rotates without slipping or stretching. Compute the length of the path that the dot travels before it returns to its initial position at the top corner.
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(10));
pair A=origin,B=(48,0),C=rotate(60,A)*B;
path equi=(0,0)--(18,0)--(9,9*sqrt(3))--cycle,circ=circle(centroid(A,B,C)*18/48,1/3);
picture a;
fill(a,equi,grey);
fill(a,circ,white);
add(a);
add(shift(15,15*sqrt(3))*a);
add(shift(30,0)*a);
draw(A--B--C--cycle,linewidth(1));
path top = circle(C,2/3);
unfill(top);
draw(top);
real r=-5/2;
draw((9,r+1)--(9,r-1)^^(9,r)--(39,r)^^(39,r-1)--(39,r+1));
label("$30$",(24,r),S);
[/asy]
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
$ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $
Let $K$ be a point on the side $[AB]$, and $L$ be a point on the side $[BC]$ of the square $ABCD$. If $|AK|=3$, $|KB|=2$, and the distance of $K$ to the line $DL$ is $3$, what is $|BL|:|LC|$?
$ \textbf{(A)}\ \frac78
\qquad\textbf{(B)}\ \frac{\sqrt 3}2
\qquad\textbf{(C)}\ \frac 87
\qquad\textbf{(D)}\ \frac 38
\qquad\textbf{(E)}\ \frac{\sqrt 2}2
$
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
[asy]
scale(200);
draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle));
path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180);
draw(p);
p=rotate(90)*p; draw(p);
p=rotate(90)*p; draw(p);
p=rotate(90)*p; draw(p);
draw(scale((sqrt(5)-1)/4)*unitcircle);
[/asy]
$\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3$
Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.
The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy]
size(200);
defaultpen(linewidth(0.7));
path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin;
path laceR=reflect((75,0),(75,-240))*laceL;
draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray);
for(int i=0;i<=3;i=i+1)
{
path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5);
unfill(circ1); draw(circ1);
unfill(circ2); draw(circ2);
}
draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]
Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.
Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 14$
Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$.
[i]Proposed by David Altizio[/i]
Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.
Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a+b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$
The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters?
[asy]/* AMC8 2003 #21 Problem */
size(4inch,2inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);[/asy]
$ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$
Consider a pyramid $P-ABCD$ whose base $ABCD$ is a square and whose vertex $P$ is equidistant from $A$, $B$, $C$, and $D$. If $AB=1$ and $\angle APD=2\theta$ then the volume of the pyramid is
$\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}$
In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2.5cm);
real m = 0;
real b = 0;
pair O = origin;
pair X = (-1,0);
pair Y = (1,0);
pair Q = midpoint(O--X);
pair A = (Q.x, -1*sqrt(3)/2);
pair B = (Q.x, -1*A.y);
pair M = (Q.x + sqrt(3)/2,0);
m = (B.y - M.y)/(B.x - M.x);
b = (B.y - m*B.x);
pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
m = (A.y - M.y)/(A.x - M.x);
b = (A.y - m*A.x);
pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
draw(Circle(O,1));
draw(Arc(Q,sqrt(3)/2,-90,90));
draw(A--B);
draw(X--Y);
draw(B--D);
draw(A--C);
draw(A--D);
dot(O);dot(M);
label("$B$",B,NW);
label("$C$",C,NE);
label("$Y$",Y,E);
label("$D$",D,SE);
label("$A$",A,SW);
label("$X$",X,W);
label("$Q$",Q,SW);
label("$O$",O,SW);
label("$M$",M,NE+2N);[/asy]$ \textbf{(A)}\ r\sqrt {2} \qquad \textbf{(B)}\ r\qquad \textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad \textbf{(D)}\ \frac {r\sqrt {3}}{2}\qquad \textbf{(E)}\ r\sqrt {3}$
During the van ride from the Grand Canyon to the beach, Michael asks his dad about the costs of renewable energy resources. "How much more does it really cost for a family like ours to switch entirely to renewable energy?"
Jerry explains, "Part of that depends on where the family lives. In the Western states, solar energy pays off more than it does where we live in the Southeast. But as technology gets better, costs of producing more photovoltaic power go down, so in just a few years more people will have reasonably inexpensive options for switching to clearner power sources. Even now most families could switch to biomass for between $\$200$ and $\$1000$ per year. The energy comes from sawdust, switchgrass, and even landfill gas. We pay that premium ourselves, but some families operate on a tighter budget, or don't understand the alternatives yet."
"Ew, landfill gas!" Alexis complains mockingly.
Wanting to save her own energy, Alexis decides to take a nap. She falls asleep and dreams of walking around a $2-\text{D}$ coordinate grid, looking for a wormhole that she believes will transport her to the beach (bypassing the time spent in the family van). In her dream, Alexis finds herself holding a device that she recognizes as a $\textit{tricorder}$ from one of the old $\textit{Star Trek}$ t.v. series. The tricorder has a button labeled "wormhole" and when Alexis presses the button, a computerized voice from the tricorder announces, "You are at the origin. Distance to the wormhole is $2400$ units. Your wormhole distance allotment is $\textit{two}$."'
Unsure as to how to reach, Alexis begins walking forward. As she walks, the tricorder displays at all times her distance from her starting point at the origin. When Alexis is $2400$ units from the origin, she again presses the "wormhole" buttom. The same computerized voice as before begins, "Distance to the origin is $2400$ units. Distance to the wormhole is $3840$ units. Your wormhole distance allotment is $\textit{two}$."
Alexis begins to feel disoriented. She wonders what is means that her $\textit{wormhole distance allotment is two}$, and why that number didn't change as she pushed the button. She puts her hat down to mark her position, then wanders aroud a bit. The tricorder shows her two readings as she walks. The first she recognizes as her distance to the origin. The second reading clearly indicates her distance from the point where her hat lies - where she last pressed the button that gave her distance to the wormhole.
Alexis picks up her hat and begins walking around. Eventually Alexis finds herself at a spot $2400$ units from the origin and $3840$ units from where she last pressed the button. Feeling hopeful, Alexis presses the tricorder's wormhole button again. Nothing happens. She presses it again, and again nothing happens. "Oh," she thinks, "my wormhole allotment was $\textit{two}$, and I used it up already!"
Despair fills poor Alexis who isn't sure what a wormhole looks like or how she's supposed to find it. Then she takes matters into her own hands. Alexis sits down and scribbles some notes and realizes where the wormhole must be. Alexis gets up and runs straight from her "third position" to the wormhole. As she gets closer, she sees the wormhole, which looks oddly like a huge scoop of icecream. Alexis runs into the wormhole, then wakes up.
How many units did Alexis run from her third position to the wormhole?
Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is
[asy]
// note: diagram deliberately not to scale -- azjps
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);
real min = -0.6, step = 0.5;
pair[] A, B; D(unitcircle);
for(int i = 0; i < 3; ++i) {
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);
D(D(A[i])--D(B[i]));
}
MP("10",(A[0]+B[0])/2,N);
MP("\sqrt{a}",(A[1]+B[1])/2,N);
MP("14",(A[2]+B[2])/2,N);
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy]
$\textbf{(A)}\ 144 \qquad
\textbf{(B)}\ 156 \qquad
\textbf{(C)}\ 168 \qquad
\textbf{(D)}\ 176 \qquad
\textbf{(E)}\ 184$
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.