MathDB

22

Part of 2002 AMC 12/AHSME

Problems(2)

Segment Length Probability

Source:

4/6/2008
Triangle ABC ABC is a right triangle with ∠ACB \angle ACB as its right angle, m\angle ABC \equal{} 60^\circ, and AB \equal{} 10. Let P P be randomly chosen inside △ABC \triangle ABC, and extend BP‾ \overline{BP} to meet AC‾ \overline{AC} at D D. What is the probability that BD>52 BD > 5\sqrt2? [asy]import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P);
draw(A--C--B--cycle); draw(B--D); dot(P); label("A",A,S); label("D",D,S); label("C",C,S); label("P",P,NE); label("B",B,N);[/asy] (A)\ \frac {2 \minus{} \sqrt2}{2} \qquad (B)\ \frac {1}{3} \qquad (C)\ \frac {3 \minus{} \sqrt3}{3} \qquad (D)\ \frac {1}{2} \qquad (E)\ \frac {5 \minus{} \sqrt5}{5}
probabilitytrigonometryratiogeometryinequalities
Log Sum

Source:

1/2/2009
For all integers n n greater than 1 1, define a_n \equal{} \frac {1}{\log_n 2002}. Let b \equal{} a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 and c \equal{} a_{10} \plus{} a_{11} \plus{} a_{12} \plus{} a_{13} \plus{} a_{14}. Then b \minus{} c equals (A)\ \minus{} 2 \qquad (B)\ \minus{} 1 \qquad (C)\ \frac {1}{2002} \qquad (D)\ \frac {1}{1001} \qquad (E)\ \frac {1}{2}
logarithms