Subcontests
(25)Octahedron in a Rectangular Prism
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));
draw((0,4,3)--(5,4,3)--(5,4,0));
label("3",(5,0,3)--(5,0,0),W);
label("4",(5,0,0)--(5,4,0),S);
label("5",(5,4,0)--(0,4,0),SE);
[/asy]
<spanclass=′latex−bold′>(A)</span>1275<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>102<spanclass=′latex−bold′>(E)</span>15 Sequence
How many terms are there in the arithmetic sequence 13,16,19,…,70,73?<spanclass=′latex−bold′>(A)</span>20<spanclass=′latex−bold′>(B)</span>21<spanclass=′latex−bold′>(C)</span>24<spanclass=′latex−bold′>(D)</span>60<spanclass=′latex−bold′>(E)</span>61 Sections of a Pizza
Four siblings ordered an extra large pizza. Alex ate 51, Beth 31, and Cyril 41 of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
<spanclass=′latex−bold′>(A)</span>Alex, Beth, Cyril, Dan
<spanclass=′latex−bold′>(B)</span>Beth, Cyril, Alex, Dan
<spanclass=′latex−bold′>(C)</span>Beth, Cyril, Dan, Alex
<spanclass=′latex−bold′>(D)</span>Beth, Dan, Cyril, Alex
<spanclass=′latex−bold′>(E)</span>Dan, Beth, Cyril, Alex Staircase
Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>18<spanclass=′latex−bold′>(C)</span>20<spanclass=′latex−bold′>(D)</span>22<spanclass=′latex−bold′>(E)</span>24[asy]
size(150);
defaultpen(linewidth(0.8));
path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
for(int i=0;i<=2;i=i+1)
{
for(int j=0;j<=3-i;j=j+1)
{
filldraw(shift((i,j))*h,black);
filldraw(shift((j,i))*v,black);
}
}[/asy] Transformations of F
The letter F shown below is rotated 90∘ clockwise around the origin, then reflected in the y-axis, and then rotated a half turn around the origin. What is the final image?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2cm);
D((0,-2)--MP("y",(0,7),N));
D((-3,0)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
[/asy][asy]
import cse5;pathpen=black;pointpen=black;
unitsize(0.2cm);
D((0,-2)--MP("y",(0,7),N));
D(MP("(A) ",(-3,0),W)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
//
D((18,-2)--MP("y",(18,7),N));
D(MP("(B) ",(13,0),W)--MP("x",(21,0),E));
D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5));
//
D((36,-2)--MP("y",(36,7),N));
D(MP("(C) ",(29,0),W)--MP("x",(38,0),E));
D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3));
//
D((0,-17)--MP("y",(0,-8),N));
D(MP("(D) ",(-3,-15),W)--MP("x",(5,-15),E));
D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10));
//
D((15,-17)--MP("y",(15,-8),N));
D(MP("(E) ",(13,-15),W)--MP("x",(22,-15),E));
D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15));
[/asy] Shark's Fin Falcata
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius 3 and center (0,0) that lies in the first quadrant, the portion of the circle with radius 23 and center (0,23) that lies in the first quadrant, and the line segment from (0,0) to (3,0). What is the area of the shark's fin falcata?
[asy]
import cse5;pathpen=black;pointpen=black;
size(1.5inch);
D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W));
path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle;
draw(P,linewidth(2));
fill(P,gray);
[/asy]
<spanclass=′latex−bold′>(A)</span>54π<spanclass=′latex−bold′>(B)</span>89π<spanclass=′latex−bold′>(C)</span>34π<spanclass=′latex−bold′>(D)</span>57π<spanclass=′latex−bold′>(E)</span>23π Clockblocked
The diagram below shows the circular face of a clock with radius 20 cm and a circular disk with radius 10 cm externally tangent to the clock face at 12 o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?[asy]
size(170);
defaultpen(linewidth(0.9)+fontsize(13pt));
draw(unitcircle^^circle((0,1.5),0.5));
path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle;
for(int i=1;i<=12;i=i+1)
{
draw(0.9*dir(90-30*i)--dir(90-30*i));
label(""+(string)i+"",0.78*dir(90-30*i));
}
dot(origin);
draw(shift((0,1.87))*arrow);
draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]<spanclass=′latex−bold′>(A)</span>2 o’clock<spanclass=′latex−bold′>(B)</span>3 o’clock<spanclass=′latex−bold′>(C)</span>4 o’clock<spanclass=′latex−bold′>(D)</span>6 o’clock<spanclass=′latex−bold′>(E)</span>8 o’clock Pentagon Ratios
In the figure shown below, ABCDE is a regular pentagon and AG=1. What is FG+JH+CD?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2inch);
pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);
D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5));
D(A--C--E--B--D--cycle);
pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A);
D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle);
[/asy]
<spanclass=′latex−bold′>(A)</span>3<spanclass=′latex−bold′>(B)</span>12−45<spanclass=′latex−bold′>(C)</span>35+25<spanclass=′latex−bold′>(D)</span>1+5<spanclass=′latex−bold′>(E)</span>1011+115 Aaron the ant
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin p0=(0,0) facing to the east and walks one unit, arriving at p1=(1,0). For n=1,2,3,…, right after arriving at the point pn, if Aaron can turn 90∘ left and walk one unit to an unvisited point pn+1, he does that. Otherwise, he walks one unit straight ahead to reach pn+1. Thus the sequence of points continues p2=(1,1),p3=(0,1),p4=(−1,1),p5=(−1,0), and so on in a counterclockwise spiral pattern. What is p2015?<spanclass=′latex−bold′>(A)</span>(−22,−13)<spanclass=′latex−bold′>(B)</span>(−13,−22)<spanclass=′latex−bold′>(C)</span>(−13,22)<spanclass=′latex−bold′>(D)</span>(13,−22)<spanclass=′latex−bold′>(E)</span>(22,−13) Volume and Surface area
A rectangular box measures a×b×c, where a, b, and c are integers and 1≤a≤b≤c. The volume and surface area of the box are numerically equal. How many ordered triples (a,b,c) are possible?<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>21<spanclass=′latex−bold′>(E)</span>26