Subcontests
(25)AMC 8 2004 Problem 23
Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?[asy]pair J=(0,6), K=origin, L=(10,0), M=(10,6);
draw(J--K--L--M--cycle);
label("J", J, dir((5,3)--J));
label("K", K, dir((5,3)--K));
label("L", L, dir((5,3)--L));
label("M", M, dir((5,3)--M));[/asy]<spanclass=′latex−bold′>(A)</span>
[asy]size(80);defaultpen(linewidth(0.8));
draw((16,0)--origin--(0,16));
draw(origin--(15,15));
label("time", (8,0), S);
label(rotate(90)*"distance", (0,8), W);
[/asy]
<spanclass=′latex−bold′>(B)</span>
[asy]size(80);defaultpen(linewidth(0.8));
draw((16,0)--origin--(0,16));
draw((0,6)--(1,6)--(1,12)--(2,12)--(2,11)--(3,11)--(3,1)--(12,1)--(12,0));
label("time", (8,0), S);
label(rotate(90)*"distance", (0,8), W);
[/asy]
<spanclass=′latex−bold′>(C)</span>
[asy]size(80);defaultpen(linewidth(0.8));
draw((16,0)--origin--(0,16));
draw(origin--(2.7,8)--(3,9)^^(11,9)--(14,0));
draw(Arc((4,9), 1, 0, 180));
draw(Arc((10,9), 1, 0, 180));
draw(Arc((7,9), 2, 180,360));
label("time", (8,0), S);
label(rotate(90)*"distance", (0,8), W);
[/asy]
<spanclass=′latex−bold′>(D)</span>
[asy]size(80);defaultpen(linewidth(0.8));
draw((16,0)--origin--(0,16));
draw(origin--(2,6)--(7,14)--(10,12)--(14,0));
label("time", (8,0), S);
label(rotate(90)*"distance", (0,8), W);
[/asy]
<spanclass=′latex−bold′>(E)</span>
[asy]size(80);defaultpen(linewidth(0.8));
draw((16,0)--origin--(0,16));
draw(origin--(3,6)--(7,6)--(10,12)--(14,12));
label("time", (8,0), S);
label(rotate(90)*"distance", (0,8), W);
[/asy] AMC 8 2004 Problem 25
Two 4×4 squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?[asy]
filldraw((0,1)--(1,2)--(3,0)--(1,-2)--(0,-1)--(-1,-2)--(-3,0)--(-1,2)--cycle, gray, black+linewidth(0.8));
filldraw(Circle(origin, 1.01), white, black+linewidth(0.8));
[/asy]<spanclass=′latex−bold′>(A)</span> 16−4π<spanclass=′latex−bold′>(B)</span> 16−2π<spanclass=′latex−bold′>(C)</span> 28−4π<spanclass=′latex−bold′>(D)</span> 28−2π<spanclass=′latex−bold′>(E)</span> 32−2π AMC 8 2004 Problem 24
In the figure, ABCD is a rectangle and EFGH is a parallelogram. Using the measurements given in the figure, what is the length d of the segment that is perpendicular to HE and FG?[asy]
defaultpen(linewidth(0.8));
size(200);
pair A=(0,8), B=(10,8), C=(10,0), D=origin;
pair E=(4,8), F=(10,3), G=(6,0), H=(0,5);
pair I=H+4*dir(H--E);
pair J=foot(I, F, G);draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
draw(I--J);
draw(rightanglemark(H,I,J));
draw(rightanglemark(F,J,I));label("A", A, dir((5,4)--A));
label("B", B, dir((5,4)--B));
label("C", C, dir((5,4)--C));
label("D", D, dir((5,4)--D));
label("E", E, dir((5,4)--E));
label("F", F, dir((5,4)--F));
label("G", G, dir((5,4)--G));
label("H", H, dir((5,4)--H));
label("d", I--J, SW);label("3", H--A, W);
label("4", E--A, N);
label("6", E--B, N);
label("5", F--B, dir(1));
label("3", F--C, dir(1));
label("5", H--D, W);
label("4", C--G, S);
label("6", D--G, S);
[/asy]
<spanclass=′latex−bold′>(A)</span> 6.8<spanclass=′latex−bold′>(B)</span> 7.1<spanclass=′latex−bold′>(C)</span> 7.6<spanclass=′latex−bold′>(D)</span> 7.8<spanclass=′latex−bold′>(E)</span> 8.1 AMC 8 2004 Problem 21
Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?[asy]
defaultpen(linewidth(1));
draw(unitcircle);
draw((1,0)--(-1,0));
draw((0,1)--(0,-1));
draw(shift(3,0)*unitcircle);
draw(shift(3,0)*(origin--dir(90)));
draw(shift(3,0)*(origin--dir(210)));
draw(shift(3,0)*(origin--dir(330)));draw(0.7*dir(200)--0.7*dir(20), linewidth(0.7), EndArrow(7));
draw(shift(3,0)*(0.7*dir(180+65)--0.7*dir(65)), linewidth(0.7), EndArrow(7));label("1", (-0.45,0.1), N);
label("4", (-0.45,-0.1), S);
label("3", (0.45,-0.1), S);
label("2", (0.45,0.1), N);label("1", shift(3,0)*(-0.25,0.1), NW);
label("2", shift(3,0)*(0.25,0.1), NE);
label("3", shift(3,0)*(0,-0.3), S);label("A", (0,-1), S);
label("B", (3,-1), S);
[/asy]<spanclass=′latex−bold′>(A)</span> 41<spanclass=′latex−bold′>(B)</span> 31<spanclass=′latex−bold′>(C)</span> 21<spanclass=′latex−bold′>(D)</span> 32<spanclass=′latex−bold′>(E)</span> 43 AMC 8 2004 Problem 15
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?[asy]
defaultpen(linewidth(1));
real x=sqrt(3)/2;
path p=rotate(30)*polygon(6);
filldraw(p^^shift(0,3)*p^^shift(4x,0)*p^^shift(3x,1.5)*p^^shift(2x,3)*p^^shift(-4x,0)*p^^shift(-3x,1.5)*p^^shift(-2x,3)*p^^shift(3x,-1.5)*p^^shift(-3x,-1.5)*p^^shift(2x,-3)*p^^shift(-2x,-3)*p^^shift(0,-3)*p, black, black);
draw(shift(2x,0)*p^^shift(-2x,0)*p^^shift(x,1.5)*p^^shift(-x,1.5)*p^^shift(x,-1.5)*p^^shift(-x,-1.5)*p);
[/asy]<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 11<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 18 AMC 8 2004 Problem 13
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true.\begin{align*}\text{I.}&\text{ Bill is the oldest.}\\
\text{II.}&\text{ Amy is not the oldest.}\\
\text{III.}&\text{ Celine is not the youngest.}\end{align*}Rank the friends from the oldest to the youngest.<spanclass=′latex−bold′>(A)</span> Bill, Amy, Celine<spanclass=′latex−bold′>(B)</span> Amy, Bill, Celine<spanclass=′latex−bold′>(C)</span> Celine, Amy, Bill<spanclass=′latex−bold′>(D)</span> Celine, Bill, Amy<spanclass=′latex−bold′>(E)</span> Amy, Celine, Bill