Subcontests
(18)2000 AMC 12 #22
The graph below shows a portion of the curve defined by the quartic polynomial P(x) \equal{} x^4 \plus{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d. Which of the following is the smallest? (A)\ P( \minus{} 1)
<spanclass=′latex−bold′>(B)</span> The product of the zeros of P
(C)\ \text{The product of the non \minus{} real zeros of }P
<spanclass=′latex−bold′>(D)</span> The sum of the coefficients of P
<spanclass=′latex−bold′>(E)</span> The sum of the real zeros of P
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(7));size(250);
real f(real x) {
real y=1/4;
return 0.2125(x*y)^4-0.625(x*y)^3-1.6125(x*y)^2+0.325(x*y)+5.3;
}
draw(graph(f,-10.5,19.4));
draw((-13,0)--(22,0)^^(0,-10.5)--(0,15));
int i;
filldraw((-13,10.5)--(22,10.5)--(22,20)--(-13,20)--cycle,white, white);
for(i=-3; i<6; i=i+1) {
if(i!=0) {
draw((4*i,0)--(4*i,-0.2));
label(string(i), (4*i,-0.2), S);
}}
for(i=-5; i<6; i=i+1){
if(i!=0) {
draw((0,2*i)--(-0.2,2*i));
label(string(2*i), (-0.2,2*i), W);
}}
label("0", origin, SE);[/asy] Colored Octohedra
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
[asy]import three;
import math;
size(180);
defaultpen(linewidth(.8pt));
currentprojection=orthographic(2,0.2,1);triple A=(0,0,1);
triple B=(sqrt(2)/2,sqrt(2)/2,0);
triple C=(sqrt(2)/2,-sqrt(2)/2,0);
triple D=(-sqrt(2)/2,-sqrt(2)/2,0);
triple E=(-sqrt(2)/2,sqrt(2)/2,0);
triple F=(0,0,-1);
draw(A--B--E--cycle);
draw(A--C--D--cycle);
draw(F--C--B--cycle);
draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]<spanclass=′latex−bold′>(A)</span> 210<spanclass=′latex−bold′>(B)</span> 560<spanclass=′latex−bold′>(C)</span> 840<spanclass=′latex−bold′>(D)</span> 1260<spanclass=′latex−bold′>(E)</span> 1680 Circle in Circular Arcs
If circular arcs AC and BC have centers at B and A, respectively, then there exists a circle tangent to both AC⌢ and BC⌢, and to AB. If the length of BC⌢ is 12, then the circumference of the circle is
[asy]unitsize(4cm);
defaultpen(fontsize(8pt)+linewidth(.8pt));
dotfactor=3;pair O=(0,.375);
pair A=(-.5,0);
pair B=(.5,0);
pair C=shift(-.5,0)*dir(60);
draw(Arc(A,1,0,60));
draw(Arc(B,1,120,180));
draw(A--B);
draw(Circle(O,.375));
dot(A);
dot(B);
dot(C);
label("A",A,SW);
label("B",B,SE);
label("C",C,N);[/asy]<spanclass=′latex−bold′>(A)</span> 24<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 26<spanclass=′latex−bold′>(D)</span> 27<spanclass=′latex−bold′>(E)</span> 28 Circle and Bisector
A circle centered at O has radius 1 and contains the point A. Segment AB is tangent to the circle at A and \angle{AOB} \equal{} \theta. If point C lies on OA and BC bisects ∠ABO, then OC \equal{}[asy]import olympiad;
unitsize(2cm);
defaultpen(fontsize(8pt)+linewidth(.8pt));
labelmargin=0.2;
dotfactor=3;pair O=(0,0);
pair A=(1,0);
pair B=(1,1.5);
pair D=bisectorpoint(A,B,O);
pair C=extension(B,D,O,A);draw(Circle(O,1));
draw(O--A--B--cycle);
draw(B--C);label("O",O,SW);
dot(O);
label("θ",(0.1,0.05),ENE);dot(C);
label("C",C,S);dot(A);
label("A",A,E);dot(B);
label("B",B,E);[/asy] (A)\ \sec^2\theta \minus{} \tan\theta \qquad (B)\ \frac {1}{2} \qquad (C)\ \frac {\cos^2\theta}{1 \plus{} \sin\theta} \qquad (D)\ \frac {1}{1 \plus{} \sin\theta} \qquad (E)\ \frac {\sin\theta}{\cos^2\theta} 13x17 Checkerboard
A checkerboard of 13 rows and 17 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 1,2,…,17, the second row 18,19,…,34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1,2,…,13, the second column 14,15,…,26 and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).<spanclass=′latex−bold′>(A)</span> 222<spanclass=′latex−bold′>(B)</span> 333<spanclass=′latex−bold′>(C)</span> 444<spanclass=′latex−bold′>(D)</span> 555<spanclass=′latex−bold′>(E)</span> 666 List Value Progression
When the mean, median, and mode of the list
10,2,5,2,4,2,xare arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of x?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 17<spanclass=′latex−bold′>(E)</span> 20 Transformations of (1,2,3)
The point P \equal{} (1,2,3) is reflected in the xy-plane, then its image Q is rotated by 180∘ about the x-axis to produce R, and finally, R is translated by 5 units in the positive-y direction to produce S. What are the coordinates of S?
(A)\ (1,7, \minus{} 3) \qquad (B)\ ( \minus{} 1,7, \minus{} 3) \qquad (C)\ ( \minus{} 1, \minus{} 2,8) \qquad (D)\ ( \minus{} 1,3,3) \qquad (E)\ (1,3,3)