Subcontests
(19)Right Triangle Quadrilateral
Convex quadrilateral ABCD has AB=3,BC=4,CD=13,AD=12, and ∠ABC=90∘, as shown. What is the area of the quadrilateral?[asy]
unitsize(.4cm);
defaultpen(linewidth(.8pt)+fontsize(14pt));
dotfactor=2;
pair A,B,C,D;
C = (0,0);
B = (0,4);
A = (3,4);
D = (12.8,-2.8);
draw(C--B--A--D--cycle);
draw(rightanglemark(C,B,A,20));
dot("A",A,N);
dot("B",B,NW);
dot("C",C,SW);
dot("D",D,E);
[/asy]<spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 36<spanclass=′latex−bold′>(C)</span> 40<spanclass=′latex−bold′>(D)</span> 48<spanclass=′latex−bold′>(E)</span> 58.5 Triangle Algebra
Real numbers a and b are chosen with 1<a<b such that no triangle with positive area has side lengths 1,a, and b or b1,a1, and 1. What is the smallest possible value of b?<spanclass=′latex−bold′>(A)</span> 23+3<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 23+5<spanclass=′latex−bold′>(D)</span> 23+6<spanclass=′latex−bold′>(E)</span> 3 Arithmetic and Geometric Series
Let a<b<c be three integers such that a,b,c is an arithmetic progression and a,c,b is a geometric progression. What is the smallest possible value of c?<spanclass=′latex−bold′>(A)</span>−2<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>4<spanclass=′latex−bold′>(E)</span>6 Spheres in a Box
A 4×4×h rectangular box contains a sphere of radius 2 and eight smaller spheres of radius 1. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is h?[asy]
import graph3;
import solids;
real h=2+2*sqrt(7);
currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2));
currentlight=light(4,-4,4);
draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0));
draw(shift((1,3,1))*unitsphere,gray(0.85));
draw(shift((3,3,1))*unitsphere,gray(0.85));
draw(shift((3,1,1))*unitsphere,gray(0.85));
draw(shift((1,1,1))*unitsphere,gray(0.85));
draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85));
draw(shift((1,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,1,h-1))*unitsphere,gray(0.85));
draw(shift((1,1,h-1))*unitsphere,gray(0.85));
draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h));
[/asy]<spanclass=′latex−bold′>(A)</span>2+27<spanclass=′latex−bold′>(B)</span>3+25<spanclass=′latex−bold′>(C)</span>4+27<spanclass=′latex−bold′>(D)</span>45<spanclass=′latex−bold′>(E)</span>47 Square and Rectangles
In the figure, ABCD is a square of side length 1. The rectangles JKHG and EBCF are congruent. What is BE?
[asy]
unitsize(150);
pair A,B,C,D,E,F,G,H,J,K;
A=(1,0); B=(0,0); C=(0,1); D=(1,1);
draw(A--B--C--D--A);E=(2-sqrt(3),0); F=(2-sqrt(3),1);
draw(E--F);G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1);
K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0);
draw(G--H--K--J--G);label("A",A,SE);
label("B",B,SW);
label("C",C,NW);
label("D",D,NE);
label("E",E,S);
label("F",F,N);
label("G",G,E);
label("H",H,N);
label("K",K,W);
label("J",J,S);
[/asy]<spanclass=′latex−bold′>(A)</span>21(6−2)<spanclass=′latex−bold′>(B)</span>41<spanclass=′latex−bold′>(C)</span>2−3<spanclass=′latex−bold′>(D)</span>63<spanclass=′latex−bold′>(E)</span>1−22 Lattice Points on a Parabola
The parabola P has focus (0,0) and goes through the points (4,3) and (−4,−3). For how many points (x,y)∈P with integer coefficients is it true that ∣4x+3y∣≤1000?<spanclass=′latex−bold′>(A)</span>38<spanclass=′latex−bold′>(B)</span>40<spanclass=′latex−bold′>(C)</span>42<spanclass=′latex−bold′>(D)</span>44<spanclass=′latex−bold′>(E)</span>46