MathDB

4

Part of 2012 CHMMC Fall

Problems(2)

2012 Fall Team #4

Source:

3/20/2022
Consider the figure below, not drawn to scale. In this figure, assume thatAB⊥BEAB \perp BE and AD⊥DEAD \perp DE. Also, let AB=6AB = \sqrt6 and ∠BED=π6\angle BED =\frac{\pi}{6} . Find ACAC. https://cdn.artofproblemsolving.com/attachments/2/d/f87ac9f111f02e261a0b5376c766a615e8d1d8.png
geometry
2012 Fall CHMMC Tiebreaker 4 lattice points in 3D sum x'^2 <sum x^2

Source:

3/1/2024
A lattice point (x,y,z)∈Z3(x, y, z) \in Z^3 can be seen from the origin if the line from the origin does not contain any other lattice point (x′,y′,z′)(x', y', z') with (x′)2+(y′)2+(z′)2<x2+y2+z2.(x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2. Let pp be the probability that a randomly selected point on the cubic lattice Z3Z^3 can be seen from the origin. Given that 1p=∑n=i∞kns\frac{1}{p}= \sum^{\infty}_{n=i} \frac{k}{n^s} for some integers i,k i, k, and ss, find i,ki, k and ss.
combinatoricslattice pointsCHMMC