Found problems: 216
2011 Purple Comet Problems, 15
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?
1993 AIME Problems, 13
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
2005 Harvard-MIT Mathematics Tournament, 5
A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?
2003 AIME Problems, 7
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.
2024 Mozambican National MO Selection Test, P2
On a sheet divided into squares, each square measuring $2cm$, two circles are drawn such that both circles are inscribed in a square as in the figure below. Determine the minimum distance between the two circles.
1961 AMC 12/AHSME, 10
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} } $
2011 Purple Comet Problems, 10
The diagram shows a large circular dart board with four smaller shaded circles each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle, and are, therefore, tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown so that it sticks to a point randomly chosen on the dart board, then the probability that the dart sticks to a point in the shaded area is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(0.8));
filldraw(circle((0,0.5),.5),gray);
filldraw(circle((0,-0.5),.5),gray);
filldraw(circle((2/3,0),1/3),gray);
filldraw(circle((-2/3,0),1/3),gray);
draw(unitcircle);
[/asy]
2006 AMC 10, 23
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}$
1982 AMC 12/AHSME, 14
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has length
[asy]
size(250);
defaultpen(fontsize(10));
pair A=origin, O=(1,0), B=(2,0), N=(3,0), C=(4,0), P=(5,0), D=(6,0), G=tangent(A,P,1,2), E=intersectionpoints(A--G, Circle(N,1))[0], F=intersectionpoints(A--G, Circle(N,1))[1];
draw(Circle(O,1)^^Circle(N,1)^^Circle(P,1)^^G--A--D, linewidth(0.7));
dot(A^^B^^C^^D^^E^^F^^G^^O^^N^^P);
label("$A$", A, W);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$", D, dir(0));
label("$P$", P, S);
label("$N$", N, S);
label("$O$", O, S);
label("$E$", E, dir(120));
label("$F$", F, NE);
label("$G$", G, dir(100));[/asy]
$\textbf {(A) } 20 \qquad \textbf {(B) } 15\sqrt{2} \qquad \textbf {(C) } 24 \qquad \textbf{(D) } 25 \qquad \textbf {(E) } \text{none of these}$
2011 Purple Comet Problems, 14
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 $m + n + k$.
2010 Romania National Olympiad, 3
Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$.
[i]Mircea Fianu[/i]
1984 AMC 12/AHSME, 17
A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(11pt));
pair A = origin, H = (5,0), B = (13,0), C = (5,6.5);
draw(C--A--B--C--H^^rightanglemark(C,H,B,16));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$H$",H,S);
label("$15$",C/2,NW);
label("$16$",(H+B)/2,S);
[/asy]
$\textbf{(A) }120\qquad
\textbf{(B) }144\qquad
\textbf{(C) }150\qquad
\textbf{(D) }216\qquad
\textbf{(E) }144\sqrt5$
2008 USA Team Selection Test, 5
Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation
\[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0
\]
for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.
1980 AMC 12/AHSME, 23
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is
$\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$
2011 AMC 8, 20
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?
[asy]
pair A,B,C,D;
A=(3,20);
B=(35,20);
C=(47,0);
D=(0,0);
draw(A--B--C--D--cycle);
dot((0,0));
dot((3,20));
dot((35,20));
dot((47,0));
label("A",A,N);
label("B",B,N);
label("C",C,S);
label("D",D,S);
draw((19,20)--(19,0));
dot((19,20));
dot((19,0));
draw((19,3)--(22,3)--(22,0));
label("12",(21,10),E);
label("50",(19,22),N);
label("15",(1,10),W);
label("20",(41,12),E);[/asy]
$ \textbf{(A)}600\qquad\textbf{(B)}650\qquad\textbf{(C)}700\qquad\textbf{(D)}750\qquad\textbf{(E)}800 $
2006 AMC 12/AHSME, 17
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}$
1957 AMC 12/AHSME, 47
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}$
2013 Online Math Open Problems, 36
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$.
[i]Ray Li[/i]
2005 AMC 10, 10
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}$
2004 Paraguay Mathematical Olympiad, 2
Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.
1983 AIME Problems, 4
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.
[asy]
size(150); defaultpen(linewidth(0.65)+fontsize(11));
real r=10;
pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C;
path P=circle(O,r);
C=intersectionpoint(B--(B.x+r,B.y),P);
draw(Arc(O, r, 45, 360-17.0312));
draw(A--B--C);dot(A); dot(B); dot(C);
label("$A$",A,NE);
label("$B$",B,SW);
label("$C$",C,SE);
[/asy]
2005 Finnish National High School Mathematics Competition, 1
In the figure below, the centres of four squares have been connected by two line
segments. Prove that these line segments are perpendicular.
2004 AMC 10, 22
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}$
2007 AMC 10, 18
A circle of radius $ 1$ is surrounded by $ 4$ circles of radius $ r$ as shown. What is $ r$?
[asy]defaultpen(linewidth(.9pt));
real r = 1 + sqrt(2);
pair A = dir(45)*(r + 1);
pair B = dir(135)*(r + 1);
pair C = dir(-135)*(r + 1);
pair D = dir(-45)*(r + 1);
draw(Circle(origin,1));
draw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r));
draw(A--(dir(45)*r + A));
draw(B--(dir(45)*r + B));
draw(C--(dir(45)*r + C));
draw(D--(dir(45)*r + D));
draw(origin--(dir(25)));
label("$r$",midpoint(A--(dir(45)*r + A)), SE);
label("$r$",midpoint(B--(dir(45)*r + B)), SE);
label("$r$",midpoint(C--(dir(45)*r + C)), SE);
label("$r$",midpoint(D--(dir(45)*r + D)), SE);
label("$1$",origin,W);[/asy]$ \textbf{(A)}\ \sqrt {2}\qquad \textbf{(B)}\ 1 \plus{} \sqrt {2}\qquad \textbf{(C)}\ \sqrt {6}\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 2 \plus{} \sqrt {2}$
2012 AMC 10, 12
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$