Subcontests
(18)Sector Area Ratio
On circle O, points C and D are on the same side of diameter AB, \angle AOC \equal{} 30^\circ, and \angle DOB \equal{} 45^\circ. What is the ratio of the area of the smaller sector COD to the area of the circle?
[asy]unitsize(6mm);
defaultpen(linewidth(0.7)+fontsize(8pt));pair C = 3*dir (30);
pair D = 3*dir (135);
pair A = 3*dir (0);
pair B = 3*dir(180);
pair O = (0,0);
draw (Circle ((0, 0), 3));
label ("C", C, NE);
label ("D", D, NW);
label ("B", B, W);
label ("A", A, E);
label ("O", O, S);
label ("45∘", (-0.3,0.1), WNW);
label ("30∘", (0.5,0.1), ENE);
draw (A--B);
draw (O--D);
draw (O--C);[/asy]<spanclass=′latex−bold′>(A)</span> 92<spanclass=′latex−bold′>(B)</span> 41<spanclass=′latex−bold′>(C)</span> 185<spanclass=′latex−bold′>(D)</span> 247<spanclass=′latex−bold′>(E)</span> 103 Domain + Range of Transformation
A function f has domain [0,2] and range [0,1]. (The notation [a,b] denotes {x:a≤x≤b}.) What are the domain and range, respectively, of the function g defined by g(x)\equal{}1\minus{}f(x\plus{}1)?
(A)\ [\minus{}1,1],[\minus{}1,0] \qquad
(B)\ [\minus{}1,1],[0,1] \qquad
(C)\ [0,2],[\minus{}1,0] \qquad
(D)\ [1,3],[\minus{}1,0] \qquad
(E)\ [1,3],[0,1] Cube Stacking
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the 13 visible numbers have the greatest possible sum. What is that sum?
[asy]unitsize(.8cm);pen p = linewidth(.8pt);
draw(shift(-2,0)*unitsquare,p);
label("1",(-1.5,0.5));
draw(shift(-1,0)*unitsquare,p);
label("2",(-0.5,0.5));
label("32",(0.5,0.5));
draw(shift(1,0)*unitsquare,p);
label("16",(1.5,0.5));
draw(shift(0,1)*unitsquare,p);
label("4",(0.5,1.5));
draw(shift(0,-1)*unitsquare,p);
label("8",(0.5,-0.5));[/asy]<spanclass=′latex−bold′>(A)</span> 154<spanclass=′latex−bold′>(B)</span> 159<spanclass=′latex−bold′>(C)</span> 164<spanclass=′latex−bold′>(D)</span> 167<spanclass=′latex−bold′>(E)</span> 189 Heavy-Tailed Permutations
A permutation (a1,a2,a3,a4,a5) of (1,2,3,4,5) is heavy-tailed if a_1 \plus{} a_2 < a_4 \plus{} a_5. What is the number of heavy-tailed permutations?
<spanclass=′latex−bold′>(A)</span> 36<spanclass=′latex−bold′>(B)</span> 40<spanclass=′latex−bold′>(C)</span> 44<spanclass=′latex−bold′>(D)</span> 48<spanclass=′latex−bold′>(E)</span> 52 Probability of Intersecting Circles
Two circles of radius 1 are to be constructed as follows. The center of circle A is chosen uniformly and at random from the line segment joining (0,0) and (2,0). The center of circle B is chosen uniformly and at random, and independently of the first choice, from the line segment joining (0,1) to (2,1). What is the probability that circles A and B intersect?
(A) \; \frac{2\plus{}\sqrt{2}}{4} \qquad (B) \; \frac{3\sqrt{3}\plus{}2}{8} \qquad (C) \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad (D) \; \frac{2\plus{}\sqrt{3}}{4} \qquad (E) \; \frac{4 \sqrt{3} \minus{} 3}{4} Equilateral Triangles and Square Root
Let A_0\equal{}(0,0). Distinct points A1,A2,… lie on the x-axis, and distinct points B1,B2,… lie on the graph of y\equal{}\sqrt{x}. For every positive integer n, A_{n\minus{}1}B_nA_n is an equilateral triangle. What is the least n for which the length A0An≥100?
<spanclass=′latex−bold′>(A)</span> 13<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 17<spanclass=′latex−bold′>(D)</span> 19<spanclass=′latex−bold′>(E)</span> 21 Sequence of Points
A sequence (a1,b1), (a2,b2), (a3,b3), … of points in the coordinate plane satisfies (a_{n \plus{} 1}, b_{n \plus{} 1}) \equal{} (\sqrt {3}a_n \minus{} b_n, \sqrt {3}b_n \plus{} a_n)\hspace{3ex}\text{for}\hspace{3ex} n \equal{} 1,2,3,\ldots. Suppose that (a_{100},b_{100}) \equal{} (2,4). What is a_1 \plus{} b_1?<spanclass=′latex−bold′>(A)</span>minus2971<spanclass=′latex−bold′>(B)</span>minus2991<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 2981<spanclass=′latex−bold′>(E)</span> 2961