Subcontests
(25)Trefoil
The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length 2?
[asy]unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(12pt));pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180);
pair E=B+C;draw(D--E--B--O--C--B--A,linetype("4 4"));
draw(Arc(O,1,0,60),linewidth(1.2pt));
draw(Arc(O,1,120,180),linewidth(1.2pt));
draw(Arc(C,1,0,60),linewidth(1.2pt));
draw(Arc(B,1,120,180),linewidth(1.2pt));
draw(A--D,linewidth(1.2pt));
draw(O--dir(40),EndArrow(HookHead,4));
draw(O--dir(140),EndArrow(HookHead,4));
draw(C--C+dir(40),EndArrow(HookHead,4));
draw(B--B+dir(140),EndArrow(HookHead,4));label("2",O,S);
draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar);
draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar);[/asy] (A)\ \frac13\pi\plus{}\frac{\sqrt3}{2} \qquad
(B)\ \frac23\pi \qquad
(C)\ \frac23\pi\plus{}\frac{\sqrt3}{4} \qquad
(D)\ \frac23\pi\plus{}\frac{\sqrt3}{3} \qquad
(E)\ \frac23\pi\plus{}\frac{\sqrt3}{2} Ratio of Probabilities of Slips Placed in a Hat
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let p be the probability that all four slips bear the same number. Let q be the probability that two of the slips bear a number a and the other two bear a number b\not\equal{} a. What is the value of q/p?
<spanclass=′latex−bold′>(A)</span> 162<spanclass=′latex−bold′>(B)</span> 180<spanclass=′latex−bold′>(C)</span> 324<spanclass=′latex−bold′>(D)</span> 360<spanclass=′latex−bold′>(E)</span> 720 Rolling Dice with Irregular Face Digits
One fair die has faces 1, 1, 2, 2, 3, 3 and another has faces 4, 4, 5, 5, 6, 6. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?
<spanclass=′latex−bold′>(A)</span> 31<spanclass=′latex−bold′>(B)</span> 94<spanclass=′latex−bold′>(C)</span> 21<spanclass=′latex−bold′>(D)</span> 95<spanclass=′latex−bold′>(E)</span> 32 Rotating a Square
Three one-inch squares are palced with their bases on a line. The center square is lifted out and rotated 45∘, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point B from the line on which the bases of the original squares were placed?
[asy]unitsize(1inch);
defaultpen(linewidth(.8pt)+fontsize(8pt));draw((0,0)--((1/3) + 3*(1/2),0));
fill(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle, rgb(.7,.7,.7));
draw(((1/6),0)--((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6),(1/2))--cycle);
draw(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle);
draw(((1/6) + 1,0)--((1/6) + 1,(1/2))--((1/6) + (3/2),(1/2))--((1/6) + (3/2),0)--cycle);draw((2,0)--(2 + (1/3) + (3/2),0));
draw(((2/3) + (3/2),0)--((2/3) + 2,0)--((2/3) + 2,(1/2))--((2/3) + (3/2),(1/2))--cycle);
draw(((2/3) + (5/2),0)--((2/3) + (5/2),(1/2))--((2/3) + 3,(1/2))--((2/3) + 3,0)--cycle);label("B",((1/6) + (1/2),(1/2)),NW);
label("B",((2/3) + 2 + (1/4),(29/30)),NNE);draw(((1/6) + (1/2),(1/2)+0.05)..(1,.8)..((2/3) + 2 + (1/4)-.05,(29/30)),EndArrow(HookHead,3));fill(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle, rgb(.7,.7,.7));
draw(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle);[/asy] (A)\ 1\qquad (B)\ \sqrt {2}\qquad (C)\ \frac {3}{2}\qquad (D)\ \sqrt {2} \plus{} \frac {1}{2}\qquad (E)\ 2 Area of an Obtuse Triangle
Equilateral △ABC has side length 2, M is the midpoint of AC, and C is the midpoint of BD. What is the area of △CDM?
[asy]size(200);defaultpen(linewidth(.8pt)+fontsize(8pt));pair B = (0,0);
pair A = 2*dir(60);
pair C = (2,0);
pair D = (4,0);
pair M = midpoint(A--C);label("A",A,NW);label("B",B,SW);label("C",C, SE);label("M",M,NE);label("D",D,SE);draw(A--B--C--cycle);
draw(C--D--M--cycle);[/asy]<spanclass=′latex−bold′>(A)</span> 22<spanclass=′latex−bold′>(B)</span> 43<spanclass=′latex−bold′>(C)</span> 23<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> 2 Triangle's area ratio
Let AB be a diameter of a circle and C be a point on AB with 2 \cdot AC \equal{} BC. Let D and E be points on the circle such that DC⊥AB and DE is a second diameter. What is the ratio of the area of △DCE to the area of △ABD?
[asy]unitsize(2.5cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
pair D=dir(aCos(C.x)), E=(-D.x,-D.y);draw(A--B--D--cycle);
draw(D--E--C);
draw(unitcircle,white);
drawline(D,C);
dot(O);clip(unitcircle);
draw(unitcircle);
label("E",E,SSE);
label("B",B,E);
label("A",A,W);
label("D",D,NNW);
label("C",C,SW);draw(rightanglemark(D,C,B,2));[/asy]<spanclass=′latex−bold′>(A)</span> 61<spanclass=′latex−bold′>(B)</span> 41<spanclass=′latex−bold′>(C)</span> 31<spanclass=′latex−bold′>(D)</span> 21<spanclass=′latex−bold′>(E)</span> 32 Star and arithmetic sequence
In the five-sided star shown, the letters A,B,C,D, and E are replaced by the numbers 3,5,6,7, and 9, although not necessarily in this order. The sums of the numbers at the ends of the line segments AB,BC,CD,DE, and EA form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?[asy]
size(150);
defaultpen(linewidth(0.8));
string[] strng = {'A','D','B','E','C'};
pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234);
draw(A--B--C--D--E--cycle);
for(int i=0;i<=4;i=i+1)
{
path circ=circle(dir(90-72*i),0.125);
unfill(circ);
draw(circ);
label(""+strng+"",dir(90-72*i));
}
[/asy]<spanclass=′latex−bold′>(A)</span> 9<spanclass=′latex−bold′>(B)</span> 10<spanclass=′latex−bold′>(C)</span> 11<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 13 Square and inner square
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 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]<spanclass=′latex−bold′>(A)</span> 25<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 36<spanclass=′latex−bold′>(D)</span> 40<spanclass=′latex−bold′>(E)</span> 42 Floor Tile Pattern
An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));fill(unitsquare,gray);
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);[/asy] (A)\ 80\minus{}20\pi \qquad
(B)\ 60\minus{}10\pi \qquad
(C)\ 80\minus{}10\pi \qquad
(D)\ 60\plus{}10\pi \qquad
(E)\ 80\plus{}10\pi