Subcontests
(25)2011 AMC 8 - Problem 9 - Carmen riding a bike
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?[asy]
import graph; size(8.76cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.58,xmax=10.19,ymin=-4.43,ymax=9.63;
draw((0,0)--(0,8)); draw((0,0)--(8,0)); draw((0,1)--(8,1)); draw((0,2)--(8,2)); draw((0,3)--(8,3)); draw((0,4)--(8,4)); draw((0,5)--(8,5)); draw((0,6)--(8,6)); draw((0,7)--(8,7)); draw((1,0)--(1,8)); draw((2,0)--(2,8)); draw((3,0)--(3,8)); draw((4,0)--(4,8)); draw((5,0)--(5,8)); draw((6,0)--(6,8)); draw((7,0)--(7,8)); label("1",(0.95,-0.24),SE*lsf); label("2",(1.92,-0.26),SE*lsf); label("3",(2.92,-0.31),SE*lsf); label("4",(3.93,-0.26),SE*lsf); label("5",(4.92,-0.27),SE*lsf); label("6",(5.95,-0.29),SE*lsf); label("7",(6.94,-0.27),SE*lsf); label("5",(-0.49,1.22),SE*lsf); label("10",(-0.59,2.23),SE*lsf); label("15",(-0.61,3.22),SE*lsf); label("20",(-0.61,4.23),SE*lsf); label("25",(-0.59,5.22),SE*lsf); label("30",(-0.59,6.2),SE*lsf); label("35",(-0.56,7.18),SE*lsf); draw((0,0)--(1,1),linewidth(1.6)); draw((1,1)--(2,3),linewidth(1.6)); draw((2,3)--(4,4),linewidth(1.6)); draw((4,4)--(7,7),linewidth(1.6)); label("HOURS",(3.41,-0.85),SE*lsf); label("M",(-1.39,5.32),SE*lsf); label("I",(-1.34,4.93),SE*lsf); label("L",(-1.36,4.51),SE*lsf); label("E",(-1.37,4.11),SE*lsf); label("S",(-1.39,3.7),SE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>2.5<spanclass=′latex−bold′>(C)</span>4<spanclass=′latex−bold′>(D)</span>4.5<spanclass=′latex−bold′>(E)</span>5 2011 AMC 8 - Problem 7 - The bolded part
Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?
[asy]
import graph; size(7.01cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.42,xmax=14.59,ymin=-10.08,ymax=5.26;
pair A=(0,0), B=(4,0), C=(0,4), D=(4,4), F=(2,0), G=(3,0), H=(1,4), I=(2,4), J=(3,4), K=(0,-2), L=(4,-2), M=(0,-6), O=(0,-4), P=(4,-4), Q=(2,-2), R=(2,-6), T=(6,4), U=(10,0), V=(10,4), Z=(10,2), A_1=(8,4), B_1=(8,0), C_1=(6,-2), D_1=(10,-2), E_1=(6,-6), F_1=(10,-6), G_1=(6,-4), H_1=(10,-4), I_1=(8,-2), J_1=(8,-6), K_1=(8,-4);
draw(C--H--(1,0)--A--cycle,linewidth(1.6)); draw(M--O--Q--R--cycle,linewidth(1.6)); draw(A_1--V--Z--cycle,linewidth(1.6)); draw(G_1--K_1--J_1--E_1--cycle,linewidth(1.6));
draw(C--D); draw(D--B); draw(B--A); draw(A--C); draw(H--(1,0)); draw(I--F); draw(J--G); draw(C--H,linewidth(1.6)); draw(H--(1,0),linewidth(1.6)); draw((1,0)--A,linewidth(1.6)); draw(A--C,linewidth(1.6)); draw(K--L); draw((4,-6)--L); draw((4,-6)--M); draw(M--K); draw(O--P); draw(Q--R); draw(O--Q); draw(M--O,linewidth(1.6)); draw(O--Q,linewidth(1.6)); draw(Q--R,linewidth(1.6)); draw(R--M,linewidth(1.6)); draw(T--V); draw(V--U); draw(U--(6,0)); draw((6,0)--T); draw((6,2)--Z); draw(A_1--B_1); draw(A_1--Z); draw(A_1--V,linewidth(1.6)); draw(V--Z,linewidth(1.6)); draw(Z--A_1,linewidth(1.6)); draw(C_1--D_1); draw(D_1--F_1); draw(F_1--E_1); draw(E_1--C_1); draw(G_1--H_1); draw(I_1--J_1); draw(G_1--K_1,linewidth(1.6)); draw(K_1--J_1,linewidth(1.6)); draw(J_1--E_1,linewidth(1.6)); draw(E_1--G_1,linewidth(1.6));
dot(A,linewidth(1pt)+ds); dot(B,linewidth(1pt)+ds); dot(C,linewidth(1pt)+ds); dot(D,linewidth(1pt)+ds); dot((1,0),linewidth(1pt)+ds); dot(F,linewidth(1pt)+ds); dot(G,linewidth(1pt)+ds); dot(H,linewidth(1pt)+ds); dot(I,linewidth(1pt)+ds); dot(J,linewidth(1pt)+ds); dot(K,linewidth(1pt)+ds); dot(L,linewidth(1pt)+ds); dot(M,linewidth(1pt)+ds); dot((4,-6),linewidth(1pt)+ds); dot(O,linewidth(1pt)+ds); dot(P,linewidth(1pt)+ds); dot(Q,linewidth(1pt)+ds); dot(R,linewidth(1pt)+ds); dot((6,0),linewidth(1pt)+ds); dot(T,linewidth(1pt)+ds); dot(U,linewidth(1pt)+ds); dot(V,linewidth(1pt)+ds); dot((6,2),linewidth(1pt)+ds); dot(Z,linewidth(1pt)+ds); dot(A_1,linewidth(1pt)+ds); dot(B_1,linewidth(1pt)+ds); dot(C_1,linewidth(1pt)+ds); dot(D_1,linewidth(1pt)+ds); dot(E_1,linewidth(1pt)+ds); dot(F_1,linewidth(1pt)+ds); dot(G_1,linewidth(1pt)+ds); dot(H_1,linewidth(1pt)+ds); dot(I_1,linewidth(1pt)+ds); dot(J_1,linewidth(1pt)+ds); dot(K_1,linewidth(1pt)+ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]<spanclass=′latex−bold′>(A)</span>1221<spanclass=′latex−bold′>(B)</span>20<spanclass=′latex−bold′>(C)</span>25<spanclass=′latex−bold′>(D)</span>3331<spanclass=′latex−bold′>(E)</span>3721 Problem 3- 2011 AMC 8
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
[asy]
filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black);
filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black);
filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black);
draw((4,0)--(4,5));
draw((3,0)--(3,5));
draw((2,0)--(2,5));
draw((1,0)--(1,5));
draw((0,4)--(5,4));
draw((0,3)--(5,3));
draw((0,2)--(5,2));
draw((0,1)--(5,1));[/asy]<spanclass=′latex−bold′>(A)</span> 8:17<spanclass=′latex−bold′>(B)</span> 25:49<spanclass=′latex−bold′>(C)</span> 36:25<spanclass=′latex−bold′>(D)</span> 32:17<spanclass=′latex−bold′>(E)</span> 36:17 Problem 11- 2011 AMC 8
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?[asy]
size(300);
real i;
defaultpen(linewidth(0.8));
draw((0,140)--origin--(220,0));
for(i=1;i<13;i=i+1) {
draw((0,10*i)--(220,10*i));
}
label("0",origin,W);
label("20",(0,20),W);
label("40",(0,40),W);
label("60",(0,60),W);
label("80",(0,80),W);
label("100",(0,100),W);
label("120",(0,120),W);
path MonD=(20,0)--(20,60)--(30,60)--(30,0)--cycle,MonL=(30,0)--(30,70)--(40,70)--(40,0)--cycle,TuesD=(60,0)--(60,90)--(70,90)--(70,0)--cycle,TuesL=(70,0)--(70,80)--(80,80)--(80,0)--cycle,WedD=(100,0)--(100,100)--(110,100)--(110,0)--cycle,WedL=(110,0)--(110,120)--(120,120)--(120,0)--cycle,ThurD=(140,0)--(140,80)--(150,80)--(150,0)--cycle,ThurL=(150,0)--(150,110)--(160,110)--(160,0)--cycle,FriD=(180,0)--(180,70)--(190,70)--(190,0)--cycle,FriL=(190,0)--(190,50)--(200,50)--(200,0)--cycle;
fill(MonD,grey);
fill(MonL,lightgrey);
fill(TuesD,grey);
fill(TuesL,lightgrey);
fill(WedD,grey);
fill(WedL,lightgrey);
fill(ThurD,grey);
fill(ThurL,lightgrey);
fill(FriD,grey);
fill(FriL,lightgrey);
draw(MonD^^MonL^^TuesD^^TuesL^^WedD^^WedL^^ThurD^^ThurL^^FriD^^FriL);
label("M",(30,-5),S);
label("Tu",(70,-5),S);
label("W",(110,-5),S);
label("Th",(150,-5),S);
label("F",(190,-5),S);
label("M",(-25,85),W);
label("I",(-27,75),W);
label("N",(-25,65),W);
label("U",(-25,55),W);
label("T",(-25,45),W);
label("E",(-25,35),W);
label("S",(-26,25),W);[/asy]<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 12 Problem 13- 2011 AMC 8
Two congruent squares, ABCD and PQRS, have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?[asy]
filldraw((0,0)--(25,0)--(25,15)--(0,15)--cycle,white,black);
label("D",(0,0),S);
label("R",(25,0),S);
label("Q",(25,15),N);
label("A",(0,15),N);
filldraw((10,0)--(15,0)--(15,15)--(10,15)--cycle,mediumgrey,black);
label("S",(10,0),S);
label("C",(15,0),S);
label("B",(15,15),N);
label("P",(10,15),N);
[/asy]<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 24<spanclass=′latex−bold′>(E)</span> 25 2011 AMC 8- Problem 25
A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?[asy]
filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray,black);
filldraw(Circle((0,0),1), mediumgray,black);
filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]<spanclass=′latex−bold′>(A)</span> 21<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 23<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 25 Problem 19- 2011 AMC 8
How many rectangles are in this figure?[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(0,0);
B=(20,0);
C=(20,20);
D=(0,20);
draw(A--B--C--D--cycle);
E=(-10,-5);
F=(13,-5);
G=(13,5);
H=(-10,5);
draw(E--F--G--H--cycle);
I=(10,-20);
J=(18,-20);
K=(18,13);
L=(10,13);
draw(I--J--K--L--cycle);[/asy]<spanclass=′latex−bold′>(A)</span> 8<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 11<spanclass=′latex−bold′>(E)</span> 12 Area Of Trapeziod
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]<spanclass=′latex−bold′>(A)</span>600<spanclass=′latex−bold′>(B)</span>650<spanclass=′latex−bold′>(C)</span>700<spanclass=′latex−bold′>(D)</span>750<spanclass=′latex−bold′>(E)</span>800 Guessing Ages
Students guess that Norb's age is 24,28,30,32,36,38,41,44,47, and 49. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?<spanclass=′latex−bold′>(A)</span>29<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>37<spanclass=′latex−bold′>(D)</span>43<spanclass=′latex−bold′>(E)</span>48 Area Relationship
Let A be the area of the triangle with sides of length 25,25, and 30. Let B be the area of the triangle with sides of length 25,25, and 40. What is the relationship between A and B?<spanclass=′latex−bold′>(A)</span>A=169B<spanclass=′latex−bold′>(B)</span>A=43B<spanclass=′latex−bold′>(C)</span>A=B<spanclass=′latex−bold′>(D)</span>A=34B<spanclass=′latex−bold′>(E)</span>A=916B Mean, Median, and Mode
Here is a list of the numbers of fish that Tyler caught in nine outings last summer: 2,0,1,3,0,3,3,1,2. Which statement about the mean, median, and mode is true?<spanclass=′latex−bold′>(A)</span>median<mean<mode<spanclass=′latex−bold′>(B)</span>mean<mode<median<spanclass=′latex−bold′>(C)</span>mean<median<mode<spanclass=′latex−bold′>(D)</span>median<mode<mean<spanclass=′latex−bold′>(E)</span>mode<median<mean