2
Part of 2000 Spain Mathematical Olympiad
Problems(2)
The probability that the persons meet [Spain 2000]
Source:
2/3/2011
The figure shows a network of roads bounding blocks. Person goes from to and person goes from to each going by a shortest path (along roads). The persons start simultaneously and go at the same constant speed. At each point with two possible directions to take, both have the same probability. Find the probability that the persons meet.[asy]
import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black;
draw((0,3)--(4,3),linewidth(1.2pt)); draw((4,3)--(4,0),linewidth(1.2pt)); draw((4,0)--(0,0),linewidth(1.2pt)); draw((0,0)--(0,3),linewidth(1.2pt)); draw((1,3)--(1,0),linewidth(1.2pt)); draw((2,3)--(2,0),linewidth(1.2pt)); draw((3,3)--(3,0),linewidth(1.2pt)); draw((0,1)--(4,1),linewidth(1.2pt)); draw((4,2)--(0,2),linewidth(1.2pt));
dot((0,0),ds); label("", (-0.3,-0.36),NE*lsf); dot((4,3),ds); label("", (4.16,3.1),NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle);
[/asy]
probabilityrectanglecombinatorics proposedcombinatoricsbinomial coefficientslattice paths
Two points with distance at most 1 [Spain 2000]
Source:
2/3/2011
Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most
trigonometrycombinatorics proposedcombinatorics