Subcontests
(25)Dart Board Probability
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));draw(A--B--C--D--E--F--G--H--cycle);
draw(A--D);
draw(B--G);
draw(C--F);
draw(E--H);[/asy]<spanclass=′latex−bold′>(A)</span> 22−1<spanclass=′latex−bold′>(B)</span> 41<spanclass=′latex−bold′>(C)</span> 22−2<spanclass=′latex−bold′>(D)</span> 42<spanclass=′latex−bold′>(E)</span> 2−2 Finding the only number that hasn't been said
Seven students count from 1 to 1000 as follows:[*]Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1,3,4,6,7,9,⋯,997,999,1000.[*]Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.[*]Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.[*]Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.[*]Finally, George says the only number that no one else says.What number does George say?<spanclass=′latex−bold′>(A)</span> 37<spanclass=′latex−bold′>(B)</span> 242<spanclass=′latex−bold′>(C)</span> 365<spanclass=′latex−bold′>(D)</span> 728<spanclass=′latex−bold′>(E)</span> 998 8 term sequence
In the eight-term sequence A,B,C,D,E,F,G,H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A+H?<spanclass=′latex−bold′>(A)</span>17<spanclass=′latex−bold′>(B)</span>18<spanclass=′latex−bold′>(C)</span>25<spanclass=′latex−bold′>(D)</span>26<spanclass=′latex−bold′>(E)</span>43 Pentagon inscribed in circle; find angle
In the given circle, the diameter EB is parallel to DC, and AB is parallel to ED. The angles AEB and ABE are in the ratio 4:5. What is the degree measure of angle BCD? [asy]
unitsize(7mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
real r=3;
pair A=(-3cos(80),-3sin(80));
pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80));
pair O=(0,0), E=(-3,0), B=(3,0);
path outer=Circle(O,r);
draw(outer);
draw(E--B);
draw(E--A);
draw(B--A);
draw(E--D);
draw(C--D);
draw(B--C);
pair[] ps={A,B,C,D,E,O};
dot(ps);
label("A",A,N);
label("B",B,NE);
label("C",C,S);
label("D",D,S);
label("E",E,NW);
label("$$",O,N);[/asy]
$ (A)\ 120 \qquad
(B)\ 125 \qquad
(C)\ 130 \qquad
(D)\ 135 \qquad
(E)\ 140 $ Ratio of elements in a set
Consider the set of numbers {1,10,102,103,...1010}. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 11<spanclass=′latex−bold′>(E)</span> 101 Area of region y=a, y=-b, x=-c, x=d
A rectangular region is bounded by the graphs of the equations y=a,y=−b,x=−c, and x=d, where a,b,c, and d are all positive numbers. Which of the following represents the area of this region? (A)\ ac+ad+bc+bd\qquad(B)\ ac-ad+bc-bd\qquad(C)\ ac+ad-bc-bd \qquad(D)\ -ac-ad+bc+bd\qquad(E)\ ac-ad-bc+bd 3-4-5 triangle; find length of segment
The area of △EBD is one third of the area of 3−4−5 △ABC. Segment DE is perpendicular to segment AB. What is BD?[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3));
pair[] ps={A,B,C,D,E};
draw(A--B--C--cycle);
draw(E--D);
draw(rightanglemark(E,D,B));
dot(ps);
label("A",A,SW);
label("B",B,SE);
label("C",C,N);
label("D",D,S);
label("E",E,NE);
label("3",midpoint(A--C),NW);
label("4",midpoint(C--B),NE);
label("5",midpoint(A--B),SW);[/asy]<spanclass=′latex−bold′>(A)</span> 34<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 49<spanclass=′latex−bold′>(D)</span> 343<spanclass=′latex−bold′>(E)</span> 25 Evaluating averages of [a,b] and {a,b,c}
Suppose [ab] denotes the average of a and b, and {abc} denotes the average of a, b, and c. What is {{110}[01]0}?<spanclass=′latex−bold′>(A)</span> 92<spanclass=′latex−bold′>(B)</span> 185<spanclass=′latex−bold′>(C)</span> 31<spanclass=′latex−bold′>(D)</span> 187<spanclass=′latex−bold′>(E)</span> 32 Perimeter of last triangle in a sequence
Let T1 be a triangle with sides 2011,2012, and 2013. For n≥1, if Tn=△ABC and D,E, and F are the points of tangency of the incircle of △ABC to the sides AB,BC and AC, respectively, then Tn+1 is a triangle with side lengths AD,BE, and CF, if it exists. What is the perimeter of the last triangle in the sequence (Tn)?<spanclass=′latex−bold′>(A)</span> 81509<spanclass=′latex−bold′>(B)</span> 321509<spanclass=′latex−bold′>(C)</span> 641509<spanclass=′latex−bold′>(D)</span> 1281509<spanclass=′latex−bold′>(E)</span> 2561509 Pairwise differences of numbers
Brian writes down four integers w>x>y>z whose sum is 44. The pairwise positive differences of these numbers are 1,3,4,5,6, and 9. What is the sum of the possible values for w?<spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 31<spanclass=′latex−bold′>(C)</span> 48<spanclass=′latex−bold′>(D)</span> 62<spanclass=′latex−bold′>(E)</span> 93 Area in circle c, outside circle a and circle b
Circles A,B, and C each have radius 1. Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside circle A and circle B ?[asy]
pathpen = linewidth(.7); pointpen = black;
pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1));[/asy]<spanclass=′latex−bold′>(A)</span> 3−2π<spanclass=′latex−bold′>(B)</span> 2π<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 43π<spanclass=′latex−bold′>(E)</span> 1+2π Even integers from 200-700 made by the set {1,2,5,7,8,9}
How many even integers are there between 200 and 700 whose digits are all different and come from the set {1,2,5,7,8,9}?<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>20<spanclass=′latex−bold′>(C)</span>72<spanclass=′latex−bold′>(D)</span>120<spanclass=′latex−bold′>(E)</span>200