Subcontests
(25)Cubes and Planes
In the cube ABCDEFGH with opposite vertices C and E, J and I are the midpoints of edges FB and HD, respectively. Let R be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is R2?[asy]
size(6cm);
pair A,B,C,D,EE,F,G,H,I,J;
C = (0,0);
B = (-1,1);
D = (2,0.5);
A = B+D;
G = (0,2);
F = B+G;
H = G+D;
EE = G+B+D;
I = (D+H)/2; J = (B+F)/2;
filldraw(C--I--EE--J--cycle,lightgray,black);
draw(C--D--H--EE--F--B--cycle);
draw(G--F--G--C--G--H);
draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed);
dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J);
label("A",A,E);
label("B",B,W);
label("C",C,S);
label("D",D,E);
label("E",EE,N);
label("F",F,W);
label("G",G,N);
label("H",H,E);
label("I",I,E);
label("J",J,W);
[/asy]<spanclass=′latex−bold′>(A)</span>45<spanclass=′latex−bold′>(B)</span>34<spanclass=′latex−bold′>(C)</span>23<spanclass=′latex−bold′>(D)</span>1625<spanclass=′latex−bold′>(E)</span>49 Squares and Diagonals
Point E is the midpoint of side CD in square ABCD, and BE meets diagonal AC at F. The area of quadrilateral AFED is 45. What is the area of ABCD?[asy]
size(5cm);
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((0,6)--(6,0)); draw((3,0)--(6,6));
label("A",(0,6),NW);
label("B",(6,6),NE);
label("C",(6,0),SE);
label("D",(0,0),SW);
label("E",(3,0),S);
label("F",(4,2),E);
[/asy]<spanclass=′latex−bold′>(A)</span>100<spanclass=′latex−bold′>(B)</span>108<spanclass=′latex−bold′>(C)</span>120<spanclass=′latex−bold′>(D)</span>135<spanclass=′latex−bold′>(E)</span>144 Similar Triangles
In △ABC, a point E is on AB with AE=1 and EB=2. Point D is on AC so that DE∥BC and point F is on BC so that EF∥AC. What is the ratio of the area of CDEF to the area of △ABC?[asy]
size(7cm);
pair A,B,C,DD,EE,FF;
A = (0,0); B = (3,0); C = (0.5,2.5);
EE = (1,0);
DD = intersectionpoint(A--C,EE--EE+(C-B));
FF = intersectionpoint(B--C,EE--EE+(C-A));
draw(A--B--C--A--DD--EE--FF,black+1bp);
label("A",A,S); label("B",B,S); label("C",C,N);
label("D",DD,W); label("E",EE,S); label("F",FF,NE);
label("1",(A+EE)/2,S); label("2",(EE+B)/2,S);
[/asy]<spanclass=′latex−bold′>(A)</span>94<spanclass=′latex−bold′>(B)</span>21<spanclass=′latex−bold′>(C)</span>95<spanclass=′latex−bold′>(D)</span>53<spanclass=′latex−bold′>(E)</span>32 Plus Minus Pyramids
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?[asy]
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("+",(0,0));
draw(shift(1,0)*box); label("−",(1,0));
draw(shift(2,0)*box); label("+",(2,0));
draw(shift(3,0)*box); label("−",(3,0));
draw(shift(0.5,0.4)*box); label("−",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("−",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("−",(2.5,0.4));
draw(shift(1,0.8)*box); label("+",(1,0.8));
draw(shift(2,0.8)*box); label("+",(2,0.8));
draw(shift(1.5,1.2)*box); label("+",(1.5,1.2));
[/asy]<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>8<spanclass=′latex−bold′>(D)</span>12<spanclass=′latex−bold′>(E)</span>16 Circles in Circles
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?[asy]
size(4cm);
filldraw(scale(2)*unitcircle,gray,black);
filldraw(shift(-1,0)*unitcircle,white,black);
filldraw(shift(1,0)*unitcircle,white,black);
[/asy]<spanclass=′latex−bold′>(A)</span>41<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>21<spanclass=′latex−bold′>(D)</span>1<spanclass=′latex−bold′>(E)</span>2π Mr. Garcia
Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.[asy]
size(8cm);
void drawbar(real x, real h) {
fill((x-0.15,0.5)--(x+0.15,0.5)--(x+0.15,h)--(x-0.15,h)--cycle,gray);
}
draw((0.5,0.5)--(7.5,0.5)--(7.5,5)--(0.5,5)--cycle);
for (real i=1; i<5; i=i+0.5) {
draw((0.5,i)--(7.5,i),gray);
}
drawbar(1.0,1.0);
drawbar(2.0,2.0);
drawbar(3.0,1.5);
drawbar(4.0,3.5);
drawbar(5.0,4.5);
drawbar(6.0,2.0);
drawbar(7.0,1.5);
for (int i=1; i<8; ++i) {
label(""+string(i)+"",(i,0.25));
}
for (int i=1; i<9; ++i) {
label(""+string(i)+"",(0.5,0.5*(i+1)),W);
}
label("Number of Days of Exercise",(4,-0.1));
label(rotate(90)*"Number of Students",(-0.1,2.75));
[/asy]What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?<spanclass=′latex−bold′>(A)</span>3.50<spanclass=′latex−bold′>(B)</span>3.57<spanclass=′latex−bold′>(C)</span>4.36<spanclass=′latex−bold′>(D)</span>4.50<spanclass=′latex−bold′>(E)</span>5.00 Twelve Sides
The twelve-sided figure shown has been drawn on 1 cm×1 cm graph paper. What is the area of the figure in cm2?[asy]
unitsize(8mm);
for (int i=0; i<7; ++i) {
draw((i,0)--(i,7),gray);
draw((0,i+1)--(7,i+1),gray);
}
draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp);
[/asy]<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>12.5<spanclass=′latex−bold′>(C)</span>13<spanclass=′latex−bold′>(D)</span>13.5<spanclass=′latex−bold′>(E)</span>14