Subcontests
(18)Unit Cube and Octahedron
A unit cube has vertices P1,P2,P3,P4,P1′,P2′,P3′, and P4′. Vertices P2,P3, and P4 are adjacent to P1, and for 1≤i≤4, vertices Pi and Pi′ are opposite to each other. A regular octahedron has one vertex in each of the segments P1P2,P1P3,P1P4,P1′P2′,P1′P3′, and P1′P4′. What is the octahedron's side length?
[asy]
import three;size(7.5cm);
triple eye = (-4, -8, 3);
currentprojection = perspective(eye);triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience
triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};// draw octahedron
triple pt(int k){ return (3*P[k] + P[1])/4; }
triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; }
draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6));
draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6));
draw(ptp(2)--pt(4), gray(0.6));
draw(pt(2)--ptp(4), gray(0.6));
draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4"));
draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));// draw cube
for(int i = 0; i < 4; ++i){
draw(P[1]--P); draw(Pp[1]--Pp);
for(int j = 0; j < 4; ++j){
if(i == 1 || j == 1 || i == j) continue;
draw(P--Pp[j]); draw(Pp--P[j]);
}
dot(P); dot(Pp);
dot(pt(i)); dot(ptp(i));
}label("P1", P[1], dir(P[1]));
label("P2", P[2], dir(P[2]));
label("P3", P[3], dir(-45));
label("P4", P[4], dir(P[4]));
label("P1′", Pp[1], dir(Pp[1]));
label("P2′", Pp[2], dir(Pp[2]));
label("P3′", Pp[3], dir(-100));
label("P4′", Pp[4], dir(Pp[4]));
[/asy]
<spanclass=′latex−bold′>(A)</span> 432<spanclass=′latex−bold′>(B)</span> 1676<spanclass=′latex−bold′>(C)</span> 25<spanclass=′latex−bold′>(D)</span> 323<spanclass=′latex−bold′>(E)</span> 26 Red & Green lights
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light?Note: 1 foot is equal to 12 inches.<spanclass=′latex−bold′>(A)</span> 18<spanclass=′latex−bold′>(B)</span> 18.5<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 20.5<spanclass=′latex−bold′>(E)</span> 22.5 Square inscribed in Equiangular Hexagon
Square AXYZ is inscribed in equiangular hexagon ABCDEF with X on BC, Y on DE, and Z on EF. Suppose that AB=40, and EF=41(3−1). What is the side-length of the square?[asy]
size(200);
defaultpen(linewidth(1));
pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A;
draw(A--B--C--D--E--F--cycle);
draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2"));
dot("A",A,W,linewidth(4));
dot("B",B,dir(0),linewidth(4));
dot("C",C,dir(0),linewidth(4));
dot("D",D,dir(20),linewidth(4));
dot("E",E,dir(100),linewidth(4));
dot("F",F,W,linewidth(4));
dot("X",X,dir(0),linewidth(4));
dot("Y",Y,N,linewidth(4));
dot("Z",Z,W,linewidth(4));
[/asy]<spanclass=′latex−bold′>(A)</span> 293<spanclass=′latex−bold′>(B)</span> 2212+2413<spanclass=′latex−bold′>(C)</span> 203+16
<spanclass=′latex−bold′>(D)</span> 202+133<spanclass=′latex−bold′>(E)</span> 216 Set of Right Triangles
Let S={(x,y):x∈{0,1,2,3,4},y∈{0,1,2,3,4,5}, and (x,y)=(0,0)}. Let T be the set of all right triangles whose vertices are in S. For every right triangle t=△ABC with vertices A, B, and C in counter-clockwise order and right angle at A, let f(t)=tan(∠CBA). What is
t∈T∏f(t)?
[asy]
size((120));
dot((1,0));
dot((2,0));
dot((3,0));
dot((4,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((0,5));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((1,5));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((2,5));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((3,5));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
dot((4,5));
label("∘", (0,0));
label("S", (-.7,2.5));
[/asy]<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 144625<spanclass=′latex−bold′>(C)</span> 24125<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 24625 Sequences of numbers
Let {ak}k=12011 be the sequence of real numbers defined by a_1=0.201, a_2=(0.2011)^{a_1}, a_3=(0.20101)^{a_2}, a_4=(0.201011)^{a_3}, and more generally ak=⎩⎨⎧(0.k+2 digits20101⋯0101)ak−1,(0.k+2 digits20101⋯01011)ak−1,if k is odd,if k is even.Rearranging the numbers in the sequence {ak}k=12011 in decreasing order produces a new sequence {bk}k=12011. What is the sum of all the integers k, 1≤k≤2011, such that ak=bk?<spanclass=′latex−bold′>(A)</span> 671<spanclass=′latex−bold′>(B)</span> 1006<spanclass=′latex−bold′>(C)</span> 1341<spanclass=′latex−bold′>(D)</span> 2011<spanclass=′latex−bold′>(E)</span> 2012 Function Unbounded
Define the function f1 on the positive integers by setting f1(1)=1 and if n=p1e1p2e2...pkek is the prime factorization of n>1, then f1(n)=(p1+1)e1−1(p2+1)e2−1⋯(pk+1)ek−1. For every m≥2, let fm(n)=f1(fm−1(n)). For how many N in the range 1≤N≤400 is the sequence (f1(N),f2(N),f3(N),...) unbounded? Note: a sequence of positive numbers is unbounded if for every integer B, there is a member of the sequence greater than B.<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 17<spanclass=′latex−bold′>(D)</span> 18<spanclass=′latex−bold′>(E)</span> 19 Polynomials & Sum
Consider all polynomials of a complex variable, P(z)=4z4+az3+bz2+cz+d, where a,b,c and d are integers, 0≤d≤c≤b≤a≤4, and the polynomial has a zero z0 with ∣z0∣=1. What is the sum of all values P(1) over all the polynomials with these properties? <spanclass=′latex−bold′>(A)</span> 84<spanclass=′latex−bold′>(B)</span> 92<spanclass=′latex−bold′>(C)</span> 100<spanclass=′latex−bold′>(D)</span> 108<spanclass=′latex−bold′>(E)</span> 120 Trapezoid and possible areas
A trapezoid has side lengths 3,5,7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of r1n1+r2n2+r3, where r1,r2, and r3 are rational numbers and n1 and n2 are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
r1+r2+r3+n1+n2?<spanclass=′latex−bold′>(A)</span> 57<spanclass=′latex−bold′>(B)</span> 59<spanclass=′latex−bold′>(C)</span> 61<spanclass=′latex−bold′>(D)</span> 63<spanclass=′latex−bold′>(E)</span> 65 Subsets with No Pairwise Sums Divisible by Five
Let S be a subset of {1,2,3,…,30} with the property that no pair of distinct elements in S has a sum divisible by 5. What is the largest possible size of S?<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 13<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 16<spanclass=′latex−bold′>(E)</span> 18 Square in the first quadrant
Square PQRS lies in the first quadrant. Points (3,0),(5,0),(7,0), and (13,0) lie on lines SP,RQ,PQ, and SR, respectively. What is the sum of the coordinates of the center of the square PQRS?<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 6.2<spanclass=′latex−bold′>(C)</span> 6.4<spanclass=′latex−bold′>(D)</span> 6.6<spanclass=′latex−bold′>(E)</span> 6.8