Subcontests
(17)Just an average sequence
Given a finite sequence S \equal{} (a_1,a_2,\ldots,a_n) of n real numbers, let A(S) be the sequence
\left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right)
of n \minus{} 1 real numbers. Define A^1(S) \equal{} A(S) and, for each integer m, 2\le m\le n \minus{} 1, define A^m(S) \equal{} A(A^{m \minus{} 1}(S)). Suppose x>0, and let S \equal{} (1,x,x^2,\ldots,x^{100}). If A^{100}(S) \equal{} (1/2^{50}), then what is x?
(A) 1 \minus{} \frac {\sqrt {2}}2\qquad (B) \sqrt {2} \minus{} 1\qquad (C) \frac 12\qquad (D) 2 \minus{} \sqrt {2}\qquad (E) \frac {\sqrt {2}}2 Point Inside a Triangle
Isosceles △ABC has a right angle at C. Point P is inside △ABC, such that PA \equal{} 11, PB \equal{} 7, and PC \equal{} 6. Legs AC and BC have length s \equal{} \sqrt {a \plus{} b\sqrt {2}}, where a and b are positive integers. What is a \plus{} b?[asy]pointpen = black;
pathpen = linewidth(0.7);
pen f = fontsize(10);
size(5cm);
pair B = (0,sqrt(85+42*sqrt(2)));
pair A = (B.y,0);
pair C = (0,0);
pair P = IP(arc(B,7,180,360),arc(C,6,0,90));
D(A--B--C--cycle);
D(P--A);
D(P--B);
D(P--C);
MP("A",D(A),plain.E,f);
MP("B",D(B),plain.N,f);
MP("C",D(C),plain.SW,f);
MP("P",D(P),plain.NE,f);[/asy]<spanclass=′latex−bold′>(A)</span>85<spanclass=′latex−bold′>(B)</span>91<spanclass=′latex−bold′>(C)</span>108<spanclass=′latex−bold′>(D)</span>121<spanclass=′latex−bold′>(E)</span>127 Right triangle with vertices as centers of circles
The vertices of a 3 \minus{} 4 \minus{} 5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
[asy]unitsize(5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));pair B=(0,0), C=(5,0);
pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0];draw(A--B--C--cycle);
draw(Circle(C,3));
draw(Circle(A,1));
draw(Circle(B,2));label("A",A,N);
label("B",B,W);
label("C",C,E);
label("3",midpoint(B--A),NW);
label("4",midpoint(A--C),NE);
label("5",midpoint(B--C),S);[/asy]<spanclass=′latex−bold′>(A)</span>12π<spanclass=′latex−bold′>(B)</span>225π<spanclass=′latex−bold′>(C)</span>13π<spanclass=′latex−bold′>(D)</span>227π<spanclass=′latex−bold′>(E)</span>14π Tangent Circles
Circles with centers O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B are on the circle centered at O, and points C and D are on the circle centered at P, such that AD and BC are common external tangents to the circles. What is the area of hexagon AOBCPD?[asy]
unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
pair A, B, C, D;
pair[] O;
O[1] = (6,0);
O[2] = (12,0);
A = (32/6,8*sqrt(2)/6);
B = (32/6,-8*sqrt(2)/6);
C = 2*B;
D = 2*A;
draw(Circle(O[1],2));
draw(Circle(O[2],4));
draw((0.7*A)--(1.2*D));
draw((0.7*B)--(1.2*C));
draw(O[1]--O[2]);
draw(A--O[1]);
draw(B--O[1]);
draw(C--O[2]);
draw(D--O[2]);
label("A", A, NW);
label("B", B, SW);
label("C", C, SW);
label("D", D, NW);
dot("O", O[1], SE);
dot("P", O[2], SE);
label("2", (A + O[1])/2, E);
label("4", (D + O[2])/2, E);[/asy]<spanclass=′latex−bold′>(A)</span>183<spanclass=′latex−bold′>(B)</span>242<spanclass=′latex−bold′>(C)</span>36<spanclass=′latex−bold′>(D)</span>243<spanclass=′latex−bold′>(E)</span>322 Square and circle
Square ABCD has side length s, a circle centered at E has radius r, and r and s are both rational. The circle passes through D, and D lies on BE. Point F lies on the circle, on the same side of BE as A. Segment AF is tangent to the circle, and AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}. What is r/s?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=3;pair B=(0,0), C=(3,0), D=(3,3), A=(0,3);
pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6);
pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0];
pair[] dots={A,B,C,D,Ep,F};draw(A--F);
draw(Circle(Ep,5/3));
draw(A--B--C--D--cycle);dot(dots);
label("A",A,NW);
label("B",B,SW);
label("C",C,SE);
label("D",D,SW);
label("E",Ep,E);
label("F",F,NW);[/asy]<spanclass=′latex−bold′>(A)</span>21<spanclass=′latex−bold′>(B)</span>95<spanclass=′latex−bold′>(C)</span>53<spanclass=′latex−bold′>(D)</span>35<spanclass=′latex−bold′>(E)</span>59 External tangent of circles
Circles with centers (2,4) and (14,9) have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form y \equal{} mx \plus{} b with m>0. What is b?[asy]
size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label("4",(2,4)+2*expi(pi*4.5/11),(-1,0)); label("9",(14,9)+4.5*expi(pi*6/7),(1,1)); label("(2,4)",(2,4),(0.5,-1.5));label("(14,9)",(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9));[/asy]<spanclass=′latex−bold′>(A)</span>199908<spanclass=′latex−bold′>(B)</span>119909<spanclass=′latex−bold′>(C)</span>17130<spanclass=′latex−bold′>(D)</span>119911<spanclass=′latex−bold′>(E)</span>119912 Expansion
The expression
(x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006}
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
<spanclass=′latex−bold′>(A)</span>6018<spanclass=′latex−bold′>(B)</span>671,676<spanclass=′latex−bold′>(C)</span>1,007,514<spanclass=′latex−bold′>(D)</span>1,008,016<spanclass=′latex−bold′>(E)</span>2,015,028 Counting subsets
How many non-empty subsets S of {1,2,3,…,15} have the following two properties?
(1) No two consecutive integers belong to S.
(2) If S contains k elements, then S contains no number less than k.
<spanclass=′latex−bold′>(A)</span>277<spanclass=′latex−bold′>(B)</span>311<spanclass=′latex−bold′>(C)</span>376<spanclass=′latex−bold′>(D)</span>377<spanclass=′latex−bold′>(E)</span>405