Subcontests
(25)AMC 8 2008 Problem 16
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?[asy]
import three;
defaultpen(linewidth(0.8));
real r=0.5;
currentprojection=orthographic(1,1/2,1/4);
draw(unitcube, white, thick(), nolight);
draw(shift(1,0,0)*unitcube, white, thick(), nolight);
draw(shift(1,-1,0)*unitcube, white, thick(), nolight);
draw(shift(1,0,-1)*unitcube, white, thick(), nolight);
draw(shift(2,0,0)*unitcube, white, thick(), nolight);
draw(shift(1,1,0)*unitcube, white, thick(), nolight);
draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy]<spanclass=′latex−bold′>(A)</span>1:6<spanclass=′latex−bold′>(B)</span>7:36<spanclass=′latex−bold′>(C)</span>1:5<spanclass=′latex−bold′>(D)</span>7:30<spanclass=′latex−bold′>(E)</span>6:25 Volume of a wedge-AMC 8 2008 problem 21
Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
[asy]
defaultpen(linewidth(0.65));
real d=90-63.43494882;
draw(ellipse((origin), 2, 4));
fill((0,4)--(0,-4)--(-8,-4)--(-8,4)--cycle, white);
draw(ellipse((-4,0), 2, 4));
draw((0,4)--(-4,4));
draw((0,-4)--(-4,-4));
draw(shift(-2,0)*rotate(-d-5)*ellipse(origin, 1.82, 4.56), linetype("10 10"));
draw((-4,4)--(-8,4), dashed);
draw((-4,-4)--(-8,-4), dashed);
draw((-4,4.3)--(-4,5));
draw((0,4.3)--(0,5));
draw((-7,4)--(-7,-4), Arrows(5));
draw((-4,4.7)--(0,4.7), Arrows(5));
label("8 cm", (-7,0), W);
label("6 cm", (-2,4.7), N);[/asy]<spanclass=′latex−bold′>(A)</span>48<spanclass=′latex−bold′>(B)</span>75<spanclass=′latex−bold′>(C)</span>151<spanclass=′latex−bold′>(D)</span>192<spanclass=′latex−bold′>(E)</span>603 AMC 8 2008 Problem 25
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?[asy]
real d=320;
pair O=origin;
pair P=O+8*dir(d);
pair A0 = origin;
pair A1 = O+1*dir(d);
pair A2 = O+2*dir(d);
pair A3 = O+3*dir(d);
pair A4 = O+4*dir(d);
pair A5 = O+5*dir(d);
filldraw(Circle(A0, 6), white, black);
filldraw(circle(A1, 5), black, black);
filldraw(circle(A2, 4), white, black);
filldraw(circle(A3, 3), black, black);
filldraw(circle(A4, 2), white, black);
filldraw(circle(A5, 1), black, black);
[/asy]<spanclass=′latex−bold′>(A)</span> 42<spanclass=′latex−bold′>(B)</span> 44<spanclass=′latex−bold′>(C)</span> 45<spanclass=′latex−bold′>(D)</span> 46<spanclass=′latex−bold′>(E)</span> 48 AMC 8 2008 Problem 18
Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at A and ending at K. How many meters does the aardvark run?
[asy]
size((150));
draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle);
draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle);
draw((20,0)--(-20,0));
draw((0,20)--(0,-20));
draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow);
draw((-18,1)--(-12, 1), EndArrow);
draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow);
draw((1,-9)--(1,9), EndArrow);
draw((0,12)..(8.3, 8.3)..(12,0), EndArrow);
draw((12,-1)--(18,-1), EndArrow);
label("A", (0,20), N);
label("K", (20,0), E);
[/asy]
<spanclass=′latex−bold′>(A)</span> 10π+20<spanclass=′latex−bold′>(B)</span> 10π+30<spanclass=′latex−bold′>(C)</span> 10π+40<spanclass=′latex−bold′>(D)</span> 20π+20<spanclass=′latex−bold′>(E)</span> 20π+40 AMC 8 2008 Problem 8
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
[asy]
draw((0,0)--(36,0)--(36,24)--(0,24)--cycle);
draw((0,4)--(36,4));
draw((0,8)--(36,8));
draw((0,12)--(36,12));
draw((0,16)--(36,16));
draw((0,20)--(36,20));
fill((4,0)--(8,0)--(8,20)--(4,20)--cycle, black);
fill((12,0)--(16,0)--(16,12)--(12,12)--cycle, black);
fill((20,0)--(24,0)--(24,8)--(20,8)--cycle, black);
fill((28,0)--(32,0)--(32,24)--(28,24)--cycle, black);
label("\120",(0,24),W);label("$80",(0,16),W);label("$40",(0,8),W);label("Jan",(6,0),S);label("Feb",(14,0),S);label("Mar",(22,0),S);label("Apr",(30,0),S);[/asy] (A)\ 60\qquad(B)\ 70\qquad(C)\ 75\qquad(D)\ 80\qquad(E)\ 85 $ AMC 8 2008 Problem 23
In square ABCE, AF=2FE and CD=2DE. What is the ratio of the area of △BFD to the area of square ABCE?
[asy]
size((100));
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((3,0)--(9,9)--(0,3)--cycle);
dot((3,0));
dot((0,3));
dot((9,9));
dot((0,0));
dot((9,0));
dot((0,9));
label("A", (0,9), NW);
label("B", (9,9), NE);
label("C", (9,0), SE);
label("D", (3,0), S);
label("E", (0,0), SW);
label("F", (0,3), W);
[/asy]
<spanclass=′latex−bold′>(A)</span> 61<spanclass=′latex−bold′>(B)</span> 92<spanclass=′latex−bold′>(C)</span> 185<spanclass=′latex−bold′>(D)</span> 31<spanclass=′latex−bold′>(E)</span> 207 AMC 8 2008 Problem 14
Three A’s, three B’s, and three C’s are placed in the nine spaces so that each row and column contain one of each letter. If A is placed in the upper left corner, how many arrangements are possible?
[asy]
size((80));
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
draw((3,0)--(3,9));
draw((6,0)--(6,9));
draw((0,3)--(9,3));
draw((0,6)--(9,6));
label("A", (1.5,7.5));
[/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 AMC 8 2008 Problem 6
In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
[asy]
size((70));
draw((10,0)--(0,10)--(-10,0)--(0,-10)--(10,0));
draw((-2.5,-7.5)--(7.5,2.5));
draw((-5,-5)--(5,5));
draw((-7.5,-2.5)--(2.5,7.5));
draw((-7.5,2.5)--(2.5,-7.5));
draw((-5,5)--(5,-5));
draw((-2.5,7.5)--(7.5,-2.5));
fill((-10,0)--(-7.5,2.5)--(-5,0)--(-7.5,-2.5)--cycle, gray);
fill((-5,0)--(0,5)--(5,0)--(0,-5)--cycle, gray);
fill((5,0)--(7.5,2.5)--(10,0)--(7.5,-2.5)--cycle, gray);
[/asy]
<spanclass=′latex−bold′>(A)</span> 3:10<spanclass=′latex−bold′>(B)</span> 3:8<spanclass=′latex−bold′>(C)</span> 3:7<spanclass=′latex−bold′>(D)</span> 3:5<spanclass=′latex−bold′>(E)</span> 1:1 AMC 8 2008 Problem 4
In the figure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1, and the three trapezoids are congruent. What is the area of one of the trapezoids?
[asy]
size((70));
draw((0,0)--(7.5,13)--(15,0)--(0,0));
draw((1.88,3.25)--(9.45,3.25));
draw((11.2,0)--(7.5,6.5));
draw((9.4,9.7)--(5.6,3.25));
[/asy]
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7 AMC 8 2008 Problem 15
In Theresa's first 8 basketball games, she scored 7,4,3,6,8,3,1 and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?<spanclass=′latex−bold′>(A)</span> 35<spanclass=′latex−bold′>(B)</span> 40<spanclass=′latex−bold′>(C)</span> 48<spanclass=′latex−bold′>(D)</span> 56<spanclass=′latex−bold′>(E)</span> 72