MathDB

9

Part of 2008 AIME Problems

Problems(2)

Stack of Crates

Source: AIME 2008I Problem 9

3/23/2008
Ten identical crates each of dimensions 3 3 ft × \times 4 4 ft × \times 6 6 ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let mn \frac{m}{n} be the probability that the stack of crates is exactly 41 41 ft tall, where m m and n n are relatively prime positive integers. Find m m.
probabilityfunctionalgebrapolynomialnumber theoryrelatively primegeometric series
Particle Moves

Source: AIME 2008II Problem 9

4/3/2008
A particle is located on the coordinate plane at (5,0) (5,0). Define a move for the particle as a counterclockwise rotation of π/4 \pi/4 radians about the origin followed by a translation of 10 10 units in the positive x x-direction. Given that the particle's position after 150 150 moves is (p,q) (p,q), find the greatest integer less than or equal to |p|\plus{}|q|.
analytic geometrygeometrygeometric transformationrotationlinear algebramatrixAMC