MathDB

10

Part of 2014 BMT Spring

Problems(5)

BMT 2014 Spring - Geometry 10

Source:

12/29/2021
Consider 8 8 points that are a knight’s move away from the origin (i.e., the eight points {(2,1)\{(2, 1) , (2,−1)(2, -1) , (1,2)(1, 2) , (1,−2)(1, -2) , (−1,2)(-1, 2) , (−1,−2)(-1, -2) , (−2,1)(-2, 1), (−2,−1)}(-2, -1)\}). Each point has probability 12\frac12 of being visible. What is the expected value of the area of the polygon formed by points that are visible? (If exactly 0,1,20, 1, 2 points appear, this area will be zero.)
probabilitygeometry
2014 BMT Team 10

Source:

1/6/2022
A unitary divisor d of a number nn is a divisor nn that has the property gcd⁡(d,n/d)=1\gcd (d, n/d) = 1. If n=1620n = 1620, what is the sum of all of the unitary divisors of dd?
number theory
BMT 2014 Spring - Analysis 10

Source:

1/6/2022
Suppose that x3−x+10−6=0x^3-x+10^{-6}=0. Suppose that x1<x2<x3x_1<x_2<x_3 are the solutions for xx. Find the integers (a,b,c)(a,b,c) closest to 108x110^8x_1, 108x210^8x_2, and 108x310^8x_3 respectively.
Polynomials
BMT 2014 Spring - Individual 10

Source:

1/22/2022
A plane intersects a sphere of radius 1010 such that the distance from the center of the sphere to the plane is 99. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.
geometry3D geometrysphere
BMT 2014 Spring - Discrete 10

Source:

1/6/2022
Let ff be a function on (1,…,n)(1,\ldots,n) that generates a permutation of (1,…,n)(1,\ldots,n). We call a fixed point of ff any element in the original permutation such that the element's position is not changed when the permutation is applied. Given that nn is a multiple of 44, gg is a permutation whose fixed points are (1,…,n2)\left(1,\ldots,\frac n2\right), and hh is a permutation whose fixed points consist of every element in an even-numbered position. What is the expected number of fixed points in h(g(1,2,…,104))h(g(1,2,\ldots,104))?
combinatorics