23
Part of 2006 AMC 12/AHSME
Problems(2)
Just an average sequence
Source: AMC 12 A 2006 #23
2/5/2006
Given a finite sequence S \equal{} (a_1,a_2,\ldots,a_n) of real numbers, let 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 , 2\le m\le n \minus{} 1, define A^m(S) \equal{} A(A^{m \minus{} 1}(S)). Suppose , and let S \equal{} (1,x,x^2,\ldots,x^{100}). If A^{100}(S) \equal{} (1/2^{50}), then what is ?
(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
inductionPascal's TriangleAMC
Point Inside a Triangle
Source: AMC 12 2006B, Problem 23
2/17/2006
Isosceles has a right angle at . Point is inside , such that PA \equal{} 11, PB \equal{} 7, and PC \equal{} 6. Legs and have length s \equal{} \sqrt {a \plus{} b\sqrt {2}}, where and 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]
rotationgeometrygeometric transformationdilationtrigonometryAMCAIME