Subcontests
(15)Rhombus Minimization
Let ABCD be a rhombus with AC\equal{}16 and BD\equal{}30. Let N be a point on AB, and let P and Q be the feet of the perpendiculars from N to AC and BD, respectively. Which of the following is closest to the minimum possible value of PQ?
[asy]unitsize(2.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C;
pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D);draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(Np--Q);
draw(Np--P);label("D",D,SW);
label("C",C,SE);
label("B",B,NE);
label("A",A,NW);
label("N",Np,N);
label("P",P,SW);
label("Q",Q,SSE);draw(rightanglemark(Np,P,C,2));
draw(rightanglemark(Np,Q,D,2));[/asy]<spanclass=′latex−bold′>(A)</span> 6.5<spanclass=′latex−bold′>(B)</span> 6.75<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 7.25<spanclass=′latex−bold′>(E)</span> 7.5 Cubic
Part of the graph of f(x) \equal{} x^3 \plus{} bx^2 \plus{} cx \plus{} d is shown. What is b?
[asy]import graph;
unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;real y(real x)
{
return (x-1)*(x+1)*(x-2);
}path bounds=(-1.5,-1)--(1.5,-1)--(1.5,2.5)--(-1.5,2.5)--cycle;pair[] points={(-1,0),(0,2),(1,0)};
draw(bounds,white);
draw(graph(y,-1.5,1.5));
drawline((0,0),(1,0));
drawline((0,0),(0,1));
dot(points);
label("(−1,0)",(-1,0),SE);
label("(1,0)",(1,0),SW);
label("(0,2)",(0,2),NE);clip(bounds);[/asy] (A)\minus{}\!4 \qquad (B)\minus{}\!2 \qquad (C)\ 0 \qquad (D)\ 2 \qquad (E)\ 4 Random Permutation
Let S be the set of permutations of the sequence 1,2,3,4,5 for which the first term is not 1. A permutation is chosen randomly from S. The probability that the second term is 2, in lowest terms, is a/b. What is a \plus{} b?<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 11<spanclass=′latex−bold′>(D)</span> 16<spanclass=′latex−bold′>(E)</span> 19 Semicircle and Quarter Circle
Square ABCD has sides of length 4, and M is the midpoint of CD. A circle with radius 2 and center M intersects a circle with raidus 4 and center A at points P and D. What is the distance from P to AD?
[asy]unitsize(8mm);
defaultpen(linewidth(.8pt));
dotfactor=4;draw(Circle((2,0),2));
draw(Circle((0,4),4));
clip(scale(4)*unitsquare);
draw(scale(4)*unitsquare);
filldraw(Circle((2,0),0.07));
filldraw(Circle((3.2,1.6),0.07));
label("A",(0,4),NW);
label("B",(4,4),NE);
label("C",(4,0),SE);
label("D",(0,0),SW);
label("M",(2,0),S);
label("P",(3.2,1.6),N);[/asy]<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 516<spanclass=′latex−bold′>(C)</span> 413<spanclass=′latex−bold′>(D)</span> 23<spanclass=′latex−bold′>(E)</span> 27 Square Around Equilateral Triangles
Points K, L, M, and N lie in the plane of the square ABCD so that AKB, BLC, CMD, and DNA are equilateral triangles. If ABCD has an area of 16, find the area of KLMN.[asy]unitsize(2cm);
defaultpen(fontsize(8)+linewidth(0.8));
pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);
draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);
label("A",A,SE);
label("B",B,SW);
label("C",C,NW);
label("D",D,NE);
label("K",K,NNW);
label("L",L,E);
label("M",M,S);
label("N",N,W);[/asy] (A)\ 32 \qquad (B)\ 16 \plus{} 16\sqrt {3} \qquad (C)\ 48 \qquad (D)\ 32 \plus{} 16\sqrt {3} \qquad (E)\ 64 Cassandra's Watch
Cassandra sets her watch to the correct time at noon. At the actual time of 1: 00 PM, she notices that her watch reads 12: 57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10: 00 PM?
<spanclass=′latex−bold′>(A)</span> 10: 22 PM and 24 seconds<spanclass=′latex−bold′>(B)</span> 10: 24 PM<spanclass=′latex−bold′>(C)</span> 10: 25 PM
<spanclass=′latex−bold′>(D)</span> 10: 27 PM<spanclass=′latex−bold′>(E)</span> 10: 30 PM Equilateral Triangles on a Pentagon
Several figures can be made by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be constructed in this way?
[asy]unitsize(2cm);pair A=dir(306);
pair B=dir(234);
pair C=dir(162);
pair D=dir(90);
pair E=dir(18);draw(A--B--C--D--E--cycle,linewidth(.8pt));
draw(E--rotate(60,D)*E--D--rotate(60,C)*D--C--rotate(60,B)*C--B--rotate(60,A)*B--A--rotate(60,E)*A--cycle,linetype("4 4"));label("A",A,SE);
label("B",B,SW);
label("C",C,WNW);
label("D",D,N);
label("E",E,ENE);[/asy]<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> 5