A player chooses one of the numbers 1 through 4.After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides ofthe dice numbered 1 through 4. If the number chosenappears on the bottom of exactly one die after it isrolled, then the player wins1. If the number chosenappears on the bottom of both of the dice, then theplayer wins2. If the number chosen does not appearon the bottom of either of the dice, the player loses1.What is the expected return to the player, in dollars,for one roll of the dice?(A)-18(B)-116(C)0(D)116(E)182007 AMC 10 B, Problem #22—“Find out the probability of number appearing 0, 1, and2 times.”16.Difficulty:HardNCTM Standard:Data Analysis and Probability for Grades 912:Understand and apply basicconcepts of probability.Mathworld.com Classification:Probability and Statistics>Probability
A pyramid with a square base is cut by a plane that isparallel to its base and is 2 units from the base. Thesurface area of the smaller pyramid that is cut fromthe top is half the surface area of the original pyramid.What is the altitude of the original pyramid?Difficulty:HardNCTMStandard:Geometry for Grades 912:explore relationships (including congruence andsimilarity) among classes of two- and three-dimensional geometric objects, make and test conjecturesabout them, and solve problems involving them.Mathworld.com Classification:Geometry>Solid Geometry>Polyhedra>Pyramids
Letndenotethesmallestpositiveintegerthatis divisible by both 4 and 9, and whose base-10representation consists of only 4’s and 9’s, with atleast one of each. What are the last four digits ofn?