22
Part of 2003 AMC 12-AHSME
Problems(2)
Do They Meet?
Source:
2/20/2008
Objects and move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object starts at and each of its steps is either right or up, both equally likely. Object starts at and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
analytic geometryprobabilityinvariant
Rhombus Minimization
Source:
1/5/2009
Let be a rhombus with AC\equal{}16 and BD\equal{}30. Let be a point on , and let and be the feet of the perpendiculars from to and , respectively. Which of the following is closest to the minimum possible value of ?
[asy]unitsize(2.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C;
pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D);draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(Np--Q);
draw(Np--P);label("",D,SW);
label("",C,SE);
label("",B,NE);
label("",A,NW);
label("",Np,N);
label("",P,SW);
label("",Q,SSE);draw(rightanglemark(Np,P,C,2));
draw(rightanglemark(Np,Q,D,2));[/asy]
geometryrhombusinequalitiessimilar trianglesPythagorean Theorem