Subcontests
(30)A Triangle and a Point
Triangle ABC and point P in the same plane are given. Point P is equidistant from A and B, angle APB is twice angle ACB, and AC intersects BP at point D. If PB \equal{} 3 and PD \equal{} 2, then AD\cdot CD \equal{}
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 8<spanclass=′latex−bold′>(E)</span> 9
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;pair A = origin;
pair B = (2,0);
pair C = (3,1);
pair P = (1,2.25);
pair D = intersectionpoint(P--B,C--A);dot(A);dot(B);dot(C);dot(P);dot(D);
label("A",A,SW);label("B",B,SE);label("C",C,N);label("D",D,NE + N);label("P",P,N);draw(A--B--P--cycle);
draw(A--C--B--cycle);[/asy] Flattened Polyhedron
In the figure, polygons A, E, and F are isosceles right triangles; B, C, and D are squares with sides of length 1; and G is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is <spanclass=′latex−bold′>(A)</span> 1/2<spanclass=′latex−bold′>(B)</span> 2/3<spanclass=′latex−bold′>(C)</span> 3/4<spanclass=′latex−bold′>(D)</span> 5/6<spanclass=′latex−bold′>(E)</span> 4/3
[asy]
size(180);
defaultpen(linewidth(.7pt)+fontsize(10pt));draw((-1,1)--(2,1));
draw((-1,0)--(1,0));
draw((-1,1)--(-1,0));
draw((0,-1)--(0,3));
draw((1,2)--(1,0));
draw((-1,1)--(1,1));
draw((0,2)--(1,2));
draw((0,3)--(1,2));
draw((0,-1)--(2,1));
draw((0,-1)--((0,-1) + sqrt(2)*dir(-15)));
draw(((0,-1) + sqrt(2)*dir(-15))--(1,0));label("<spanclass=′latex−bold′>A</span>",foot((0,2),(0,3),(1,2)),SW);
label("<spanclass=′latex−bold′>B</span>",midpoint((0,1)--(1,2)));
label("<spanclass=′latex−bold′>C</span>",midpoint((-1,0)--(0,1)));
label("<spanclass=′latex−bold′>D</span>",midpoint((0,0)--(1,1)));
label("<spanclass=′latex−bold′>E</span>",midpoint((1,0)--(2,1)),NW);
label("<spanclass=′latex−bold′>F</span>",midpoint((0,-1)--(1,0)),NW);
label("<spanclass=′latex−bold′>G</span>",midpoint((0,-1)--(1,0)),2SE);[/asy] Sum of 100 Consecutive Integers
Which one of the following integers can be expressed as the sum of 100 consecutive positive integers?
<spanclass=′latex−bold′>(A)</span> 1,627,384,950<spanclass=′latex−bold′>(B)</span> 2,345,678,910<spanclass=′latex−bold′>(C)</span> 3,579,111,300<spanclass=′latex−bold′>(D)</span> 4,692,581,470<spanclass=′latex−bold′>(E)</span> 5,815,937,260 Circle Tangent to Special Triangle
A circle with center O is tangent to the coordinate axes and to the hypotenuse of the 30∘-60∘-90∘ triangle ABC as shown, where AB \equal{} 1. To the nearest hundredth, what is the radius of the circle?
[asy]defaultpen(linewidth(.8pt));
dotfactor=3;pair A = origin;
pair B = (1,0);
pair C = (0,sqrt(3));
pair O = (2.33,2.33);dot(A);dot(B);dot(C);dot(O);label("A",A,SW);label("B",B,SE);label("C",C,W);label("O",O,NW);
label("1",midpoint(A--B),S);label("60∘",B,2W + N);draw((3,0)--A--(0,3));
draw(B--C);
draw(Arc(O,2.33,163,288.5));[/asy]<spanclass=′latex−bold′>(A)</span> 2.18<spanclass=′latex−bold′>(B)</span> 2.24<spanclass=′latex−bold′>(C)</span> 2.31<spanclass=′latex−bold′>(D)</span> 2.37<spanclass=′latex−bold′>(E)</span> 2.41 Perpendicular Medians in a Triangle
Medians BD and CE of triangle ABC are perpendicular, BD \equal{} 8, and CE \equal{} 12. The area of triangle ABC is
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;pair A = origin;
pair B = (1.25,1);
pair C = (2,0);
pair D = midpoint(A--C);
pair E = midpoint(A--B);
pair G = intersectionpoint(E--C,B--D);dot(A);dot(B);dot(C);dot(D);dot(E);dot(G);
label("A",A,S);label("B",B,N);label("C",C,S);label("D",D,S);label("E",E,NW);label("G",G,NE);
draw(A--B--C--cycle);
draw(B--D);
draw(E--C);
draw(rightanglemark(C,G,D,3));[/asy]<spanclass=′latex−bold′>(A)</span> 24<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 48<spanclass=′latex−bold′>(D)</span> 64<spanclass=′latex−bold′>(E)</span> 96 Area of a Quadrilateral in a Square
In the figure, ABCD is a 2×2 square, E is the midpoint of AD, and F is on BE. If CF is perpendicular to BE, then the area of quadrilateral CDEF is
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;pair A = (0,2);
pair B = origin;
pair C = (2,0);
pair D = (2,2);
pair E = midpoint(A--D);
pair F = foot(C,B,E);dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);label("A",A,N);label("B",B,S);label("C",C,S);label("D",D,N);label("E",E,N);label("F",F,NW);draw(A--B--C--D--cycle);
draw(B--E);
draw(C--F);
draw(rightanglemark(B,F,C,4));[/asy] (A)\ 2\qquad (B)\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad (C)\ \frac {11}{5}\qquad (D)\ \sqrt {5}\qquad (E)\ \frac {9}{4} Perimeter of a Dissected Rectangle
A rectangle with perimeter 176 is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
[asy]defaultpen(linewidth(.8pt));
draw(origin--(0,3)--(4,3)--(4,0)--cycle);
draw((0,1)--(4,1));
draw((2,0)--midpoint((0,1)--(4,1)));
real r = 4/3;
draw((r,3)--foot((r,3),(0,1),(4,1)));
draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]<spanclass=′latex−bold′>(A)</span> 35.2<spanclass=′latex−bold′>(B)</span> 76<spanclass=′latex−bold′>(C)</span> 80<spanclass=′latex−bold′>(D)</span> 84<spanclass=′latex−bold′>(E)</span> 86 Perimeter of a Decagon
The adjacent sides of the decagon shown meet at right angles. What is its perimeter?
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;
dot(origin);dot((12,0));dot((12,1));dot((9,1));dot((9,7));dot((7,7));dot((7,10));dot((3,10));dot((3,8));dot((0,8));
draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8)--(0,8)--cycle);
label("8",midpoint(origin--(0,8)),W);
label("2",midpoint((3,8)--(3,10)),W);
label("12",midpoint(origin--(12,0)),S);[/asy]<spanclass=′latex−bold′>(A)</span> 22<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 34<spanclass=′latex−bold′>(D)</span> 44<spanclass=′latex−bold′>(E)</span> 50