Subcontests
(25)2013/8/18/Box with cover sliced off
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
[asy]
import three;
size(3inch);
currentprojection=orthographic(-8,15,15);
triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P;
A = (0,0,0);
B = (0,10,0);
C = (12,10,0);
D = (12,0,0);
E = (0,0,5);
F = (0,10,5);
G = (12,10,5);
H = (12,0,5);
I = (1,1,1);
J = (1,9,1);
K = (11,9,1);
L = (11,1,1);
M = (1,1,5);
N = (1,9,5);
O = (11,9,5);
P = (11,1,5);
//outside box far
draw(surface(A--B--C--D--cycle),white,nolight);
draw(A--B--C--D--cycle);
draw(surface(E--A--D--H--cycle),white,nolight);
draw(E--A--D--H--cycle);
draw(surface(D--C--G--H--cycle),white,nolight);
draw(D--C--G--H--cycle);
//inside box far
draw(surface(I--J--K--L--cycle),white,nolight);
draw(I--J--K--L--cycle);
draw(surface(I--L--P--M--cycle),white,nolight);
draw(I--L--P--M--cycle);
draw(surface(L--K--O--P--cycle),white,nolight);
draw(L--K--O--P--cycle);
//inside box near
draw(surface(I--J--N--M--cycle),white,nolight);
draw(I--J--N--M--cycle);
draw(surface(J--K--O--N--cycle),white,nolight);
draw(J--K--O--N--cycle);
//outside box near
draw(surface(A--B--F--E--cycle),white,nolight);
draw(A--B--F--E--cycle);
draw(surface(B--C--G--F--cycle),white,nolight);
draw(B--C--G--F--cycle);
//top
draw(surface(E--H--P--M--cycle),white,nolight);
draw(surface(E--M--N--F--cycle),white,nolight);
draw(surface(F--N--O--G--cycle),white,nolight);
draw(surface(O--G--H--P--cycle),white,nolight);
draw(M--N--O--P--cycle);
draw(E--F--G--H--cycle);
label("10",(A--B),SE);
label("12",(C--B),SW);
label("5",(F--B),W);[/asy]
<spanclass=′latex−bold′>(A)</span> 204<spanclass=′latex−bold′>(B)</span> 280<spanclass=′latex−bold′>(C)</span> 320<spanclass=′latex−bold′>(D)</span> 340<spanclass=′latex−bold′>(E)</span> 600 2013/8/19/Math test scores
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, ``I didn't get the lowest score in our class,'' and Bridget adds, ``I didn't get the highest score.'' What is the ranking of the three girls from highest to lowest?<spanclass=′latex−bold′>(A)</span> Hannah, Cassie, Bridget<spanclass=′latex−bold′>(B)</span> Hannah, Bridget, Cassie \\ <spanclass=′latex−bold′>(C)</span> Cassie, Bridget, Hannah<spanclass=′latex−bold′>(D)</span> Cassie, Hannah, Bridget \\<spanclass=′latex−bold′>(E)</span> Bridget, Cassie, Hannah 2013/8/22/Toothpicks in a grid
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?[asy]
picture corner;
draw(corner,(5,0)--(35,0));
draw(corner,(0,-5)--(0,-35));
for (int i=0; i<3; ++i)
{
for (int j=0; j>-2; --j)
{
if ((i-j)<3)
{
add(corner,(50i,50j));
}
}
}
draw((5,-100)--(45,-100));
draw((155,0)--(185,0),dotted+linewidth(2));
draw((105,-50)--(135,-50),dotted+linewidth(2));
draw((100,-55)--(100,-85),dotted+linewidth(2));
draw((55,-100)--(85,-100),dotted+linewidth(2));
draw((50,-105)--(50,-135),dotted+linewidth(2));
draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy]<spanclass=′latex−bold′>(A)</span> 1920<spanclass=′latex−bold′>(B)</span> 1952<spanclass=′latex−bold′>(C)</span> 1980<spanclass=′latex−bold′>(D)</span> 2013<spanclass=′latex−bold′>(E)</span> 3932 2013/8/23/Semicircles
Angle ABC of △ABC is a right angle. The sides of △ABC are the diameters of semicircles as shown. The area of the semicircle on AB equals 8π, and the arc of the semicircle on AC has length 8.5π. What is the radius of the semicircle on BC?
[asy]
import graph;
draw((0,8)..(-4,4)..(0,0)--(0,8));
draw((0,0)..(7.5,-7.5)..(15,0)--(0,0));
real theta = aTan(8/15);
draw(arc((15/2,4),17/2,-theta,180-theta));
draw((0,8)--(15,0));
label("A", (0,8), NW);
label("B", (0,0), SW);
label("C", (15,0), SE);[/asy]<spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 7.5<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 8.5<spanclass=′latex−bold′>(E)</span> 9 2013/8/24/Three squares
Squares ABCD, EFGH, and GHIJ are equal in area. Points C and D are the midpoints of sides IH ad HE, respectively. What is the ratio of the area of the shaded pentagon AJICB to the sum of the areas of the three squares?[asy]
pair A,B,C,D,E,F,G,H,I,J;
A = (0.5,2);
B = (1.5,2);
C = (1.5,1);
D = (0.5,1);
E = (0,1);
F = (0,0);
G = (1,0);
H = (1,1);
I = (2,1);
J = (2,0);
draw(A--B);
draw(C--B);
draw(D--A);
draw(F--E);
draw(I--J);
draw(J--F);
draw(G--H);
draw(A--J);
filldraw(A--B--C--I--J--cycle,grey);
draw(E--I);
dot("A", A, NW);
dot("B", B, NE);
dot("C", C, NE);
dot("D", D, NW);
dot("E", E, NW);
dot("F", F, SW);
dot("G", G, S);
dot("H", H, N);
dot("I", I, NE);
dot("J", J, SE);[/asy]<spanclass=′latex−bold′>(A)</span> 41<spanclass=′latex−bold′>(B)</span> 247<spanclass=′latex−bold′>(C)</span> 31<spanclass=′latex−bold′>(D)</span> 83<spanclass=′latex−bold′>(E)</span> 125 2013/8/25/Rolling on a semicircle
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are R1=100 inches, R2=60 inches, and R3=80 inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
[asy]
size(8cm);
draw((0,0)--(480,0),linetype("3 4"));
filldraw(circle((8,0),8),black);
draw((0,0)..(100,-100)..(200,0));
draw((200,0)..(260,60)..(320,0));
draw((320,0)..(400,-80)..(480,0));
draw((100,0)--(150,-50sqrt(3)),Arrow(size=4));
draw((260,0)--(290,30sqrt(3)),Arrow(size=4));
draw((400,0)--(440,-40sqrt(3)),Arrow(size=4));
label("R1",(100,0)--(150,-50sqrt(3)), W, fontsize(10));
label("R2",(260,0)--(290,30sqrt(3)), W, fontsize(10));
label("R3",(400,0)--(440,-40sqrt(3)), W, fontsize(10));
filldraw(circle((8,0),8),black);
label("A",(0,0),W,fontsize(10));[/asy]<spanclass=′latex−bold′>(A)</span> 238π<spanclass=′latex−bold′>(B)</span> 240π<spanclass=′latex−bold′>(C)</span> 260π<spanclass=′latex−bold′>(D)</span> 280π<spanclass=′latex−bold′>(E)</span> 500π 2013/8/5/Mean vs. median
Hammie is in the 6th grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?<spanclass=′latex−bold′>(A)</span> median, by 60<spanclass=′latex−bold′>(B)</span> median, by 20<spanclass=′latex−bold′>(C)</span> average, by 5<spanclass=′latex−bold′>(D)</span> average, by 15 \\ <spanclass=′latex−bold′>(E)</span> average, by 20 2013/8/6/Empty cell of a grid
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, 30=6×5. What is the missing number in the top row?[asy]
unitsize(0.8cm);
draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle);
draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((-3,2)--(-1,2)--(-1,4)--(-3,4)--cycle);
draw((-1,2)--(1,2)--(1,4)--(-1,4)--cycle);
draw((1,2)--(1,4)--(3,4)--(3,2)--cycle);
label("600",(0,-1));
label("30",(-1,1));
label("6",(-2,3));
label("5",(0,3));
[/asy]<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6