Subcontests
(25)Area of the Circle in the Intersection of Two Circles
A circle of radius 1 is internally tangent to two circles of radius 2 at points A and B, where AB is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles?
[asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;pair B = (0,1);
pair A = (0,-1);label("B",B,NW);label("A",A,2S);draw(Circle(A,2));draw(Circle(B,2));fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray);
fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray);
draw((-sqrt(3),0)..B..(sqrt(3),0));
draw((-sqrt(3),0)..A..(sqrt(3),0));path circ = Circle(origin,1);fill(circ,white);
draw(circ);
dot(A);dot(B);pair A1 = B + dir(45)*2;
pair A2 = dir(45);
pair A3 = dir(-135)*2 + A;draw(B--A1,EndArrow(HookHead,2));
draw(origin--A2,EndArrow(HookHead,2));
draw(A--A3,EndArrow(HookHead,2)); label("2",midpoint(B--A1),NW);
label("1",midpoint(origin--A2),NW);
label("2",midpoint(A--A3),NW);[/asy] (A)\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad (B)\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad (C)\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad(D)\ \frac {8}{3}\pi \minus{} 3\sqrt {2}
(E)\ \frac {8}{3}\pi \minus{} 2\sqrt {3} Problem: 2004 AMC 10 #23
Circles A, B, and C are externally tangent to each other and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?[asy]
size(200);
defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(Circle(origin, 2));
draw(Circle((-1,0), 1));
draw(Circle((6/9, 8/9), 8/9));
draw(Circle((6/9, -8/9), 8/9));
label("A", (-1.2, -0.2), NE);
label("B", (6/9, 7/9), N);
label("C", (6/9, -7/9), S);
label("D", 2*dir(110), dir(110));[/asy]<spanclass=′latex−bold′>(A)</span>32<spanclass=′latex−bold′>(B)</span>23<spanclass=′latex−bold′>(C)</span>87<spanclass=′latex−bold′>(D)</span>98<spanclass=′latex−bold′>(E)</span>31+3 Problem: 2004 AMC 10 #22
Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicircle from C intersects side AD at E. What is the length of CE?[asy]defaultpen(linewidth(0.8));
pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D);
draw(C--D--A--B--C--E);
draw(Arc((0.5,0), 0.5, 0, 180));
pair point=(0.5,0.5);
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("E", E, dir(point--E));[/asy]<spanclass=′latex−bold′>(A)</span>22+5<spanclass=′latex−bold′>(B)</span>5<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>25<spanclass=′latex−bold′>(E)</span>5−5 Problem: 2004 AMC 10 #21
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is 8/13 of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: π radians is 180 degrees.)[asy]
defaultpen(linewidth(0.8));
pair O=origin;
fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray);
fill(O--Arc(O, 1, 20, 160)--cycle, white);
fill(O--Arc(O, 2, 200, 340)--cycle, mediumgray);
fill(O--Arc(O, 1, 200, 340)--cycle, white);
fill(O--Arc(O, 3, 160, 200)--cycle, mediumgray);
fill(O--Arc(O, 2, 160, 200)--cycle, white);
fill(O--Arc(O, 1, 160, 200)--cycle, mediumgray);
fill(O--Arc(O, 3, -20, 20)--cycle, mediumgray);
fill(O--Arc(O, 2, -20, 20)--cycle, white);
fill(O--Arc(O, 1, -20, 20)--cycle, mediumgray);
draw(Circle(origin, 1));draw(Circle(origin, 2));draw(Circle(origin, 3));
draw(5*dir(200)--5*dir(20)^^5*dir(160)--5*dir(-20));[/asy]<spanclass=′latex−bold′>(A)</span>8π<spanclass=′latex−bold′>(B)</span>7π<spanclass=′latex−bold′>(C)</span>6π<spanclass=′latex−bold′>(D)</span>5π<spanclass=′latex−bold′>(E)</span>4π Two Arithmetic Progressions
Let 1,4,⋯ and 9,16,⋯ be two arithmetic progressions. The set S is the union of the fi rst 2004 terms of each sequence. How many distinct numbers are in S?
<spanclass=′latex−bold′>(A)</span> 3722<spanclass=′latex−bold′>(B)</span> 3732<spanclass=′latex−bold′>(C)</span> 3914<spanclass=′latex−bold′>(D)</span> 3924<spanclass=′latex−bold′>(E)</span> 4007 Problem: 2004 AMC 10 #20
Points E and F are located on square ABCD so that ΔBEF is equilateral. What is the ratio of the area of ΔDEF to that of ΔABE?[asy]
pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=B+2*dir(165), E=intersectionpoint(B--X, A--D), Y=B+2*dir(105), F=intersectionpoint(B--Y, D--C);
draw(B--C--D--A--B--F--E--B);
pair point=(0.5,0.5);
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("E", E, dir(point--E));
label("F", F, dir(point--F));[/asy]<spanclass=′latex−bold′>(A)</span>34<spanclass=′latex−bold′>(B)</span>23<spanclass=′latex−bold′>(C)</span>3<spanclass=′latex−bold′>(D)</span>2<spanclass=′latex−bold′>(E)</span>1+3 Intersecting Lines in a Triangle
In △ABC points D and E lie on BC and AC, respectively. If AD and BE intersect at T so that AT/DT \equal{} 3 and BT/ET \equal{} 4, what is CD/BD?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);label("B",B,NW);label("A",A,SW);label("C",C,SE);label("D",D,NE);label("E",E,S);label("T",T,2*W+N);draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy]<spanclass=′latex−bold′>(A)</span> 81<spanclass=′latex−bold′>(B)</span> 92<spanclass=′latex−bold′>(C)</span> 103<spanclass=′latex−bold′>(D)</span> 114<spanclass=′latex−bold′>(E)</span> 125 Problem: 2004 AMC 10 #19
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?[asy]
size(250);defaultpen(linewidth(0.8));
draw(ellipse(origin, 3, 1));
fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white);
draw((3,0)--(3,16)^^(-3,0)--(-3,16));
draw((0, 15)--(3, 12)^^(0, 16)--(3, 13));
filldraw(ellipse((0, 16), 3, 1), white, black);
draw((-3,11)--(3, 5)^^(-3,10)--(3, 4));
draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89));
draw((0,-1)--(0,15), dashed);
draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4));
draw((-7,0)--(-5,0)^^(-7,16)--(-5,16));
draw((3,-3)--(-3,-3), Arrows(6));
draw((-6,0)--(-6,16), Arrows(6));
draw((-2,9)--(-1,9), Arrows(3));
label("3", (-1.375,9.05), dir(260), fontsize(7));
label("A", (0,15), N);
label("B", (0,-1), NE);
label("30", (0, -3), S);
label("80", (-6, 8), W);[/asy]<spanclass=′latex−bold′>(A)</span>120<spanclass=′latex−bold′>(B)</span>180<spanclass=′latex−bold′>(C)</span>240<spanclass=′latex−bold′>(D)</span>360<spanclass=′latex−bold′>(E)</span>480 Recursive Sequence
In the sequence 2001,2002,2003,…, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is 2001 \plus{} 2002 \minus{} 2003 \equal{} 2000. What is the 2004th term in this sequence?
(A) \minus{} \! 2004 \qquad (B) \minus{} \! 2 \qquad (C)\ 0 \qquad (D)\ 4003 \qquad (E)\ 6007 A Right Triangle Inside Another
In right triangle △ACE, we have AC \equal{} 12, CE \equal{} 16, and EA \equal{} 20. Points B, D, and F are located on AC, CE, and EA, respectively, so that AB \equal{} 3, CD \equal{} 4, and EF \equal{} 5. What is the ratio of the area of △DBF to that of △ACE?
[asy]
size(200);defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;pair C = (0,0);
pair E = (16,0);
pair A = (0,12);
pair F = waypoint(E--A,0.25);
pair B = waypoint(A--C,0.25);
pair D = waypoint(C--E,0.25);dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);label("A",A,NW);label("B",B,W);label("C",C,SW);label("D",D,S);label("E",E,SE);label("F",F,NE);label("3",midpoint(A--B),W);
label("9",midpoint(B--C),W);
label("4",midpoint(C--D),S);
label("12",midpoint(D--E),S);
label("5",midpoint(E--F),NE);
label("15",midpoint(F--A),NE);draw(A--C--E--cycle);
draw(B--F--D--cycle);[/asy]<spanclass=′latex−bold′>(A)</span> 41<spanclass=′latex−bold′>(B)</span> 259<spanclass=′latex−bold′>(C)</span> 83<spanclass=′latex−bold′>(D)</span> 2511<spanclass=′latex−bold′>(E)</span> 167 Problem: 2004 AMC 10 #12
Henry's Hamburger Heaven orders its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?
<spanclass=′latex−bold′>(A)</span> 24<spanclass=′latex−bold′>(B)</span> 256<spanclass=′latex−bold′>(C)</span> 768<spanclass=′latex−bold′>(D)</span> 40,320<spanclass=′latex−bold′>(E)</span> 120,960 Annulus
An annulus is the region between two concentric circles. The concentric circles in the figure have radii b and c, with b>c. Let OX be a radius of the larger circle, let XZ be tangent to the smaller circle at Z, and let 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?
<spanclass=′latex−bold′>(A)</span> πa2<spanclass=′latex−bold′>(B)</span> πb2<spanclass=′latex−bold′>(C)</span> πc2<spanclass=′latex−bold′>(D)</span> πd2<spanclass=′latex−bold′>(E)</span> πe2
[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] Problem: 2004 AMC 10 #9
In the figure, ∠EAB and ∠ABC are right angles. AB \equal{} 4, BC \equal{} 6, AE \equal{} 8, and AC and BE intersect at D. What is the difference between the areas of △ADE and △BDC?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));pair A=(0,0), B=(4,0), C=(4,6), Ep=(0,8);
pair D=extension(A,C,Ep,B);draw(A--C--B--A--Ep--B);
label("A",A,SW);
label("B",B,SE);
label("C",C,N);
label("E",Ep,N);
label("D",D,2.5*N);
label("4",midpoint(A--B),S);
label("6",midpoint(B--C),E);
label("8",(0,3),W);[/asy]<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 8<spanclass=′latex−bold′>(E)</span> 9 Problem: 2004 AMC 10 #5
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
[asy]unitsize(.5cm);
defaultpen(linewidth(.8pt));
dotfactor=3;pair[] dotted={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)};dot(dotted);[/asy]<spanclass=′latex−bold′>(A)</span> 211<spanclass=′latex−bold′>(B)</span> 141<spanclass=′latex−bold′>(C)</span> 212<spanclass=′latex−bold′>(D)</span> 71<spanclass=′latex−bold′>(E)</span> 72 Problem: 2004 AMC 10 #2
For any three real numbers a, b, and c, with b=c, the operation ⊗ is defined by:
\otimes(a,b,c) \equal{} \frac {a}{b \minus{} c}
What is ⊗(⊗(1,2,3),⊗(2,3,1),⊗(3,1,2))?
(A)\ \minus{}\!\frac {1}{2}\qquad
(B)\ \minus{}\!\frac {1}{4}\qquad
(C)\ 0\qquad
(D)\ \frac {1}{4}\qquad
(E)\ \frac {1}{2}