5

Part of 2014 BMT Spring

Problems(5)

BMT 2014 Spring - Geometry 5

Source:

In a 100-dimensional hypercube, each edge has length

1

. The box contains

2^{100} + 1

hyperspheres with the same radius

r

. The center of one hypersphere is the center of the hypercube, and it touches all the other spheres. Each of the other hyperspheres is tangent to

100

faces of the hypercube. Thus, the hyperspheres are tightly packed in the hypercube. Find

r

2014 BMT Team 5

Source:

Call two regular polygons supplementary if the sum of an internal angle from each polygon adds up to

180^o

. For instance, two squares are supplementary because the sum of the internal angles is

90^o + 90^o = 180^o

. Find the other pair of supplementary polygons. Write your answer in the form

(m, n)

where m and n are the number of sides of the polygons and

m < n

geometrynumber theory

BMT 2014 Spring - Analysis 5

Source:

\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}

limitsreal analysis

BMT 2014 Spring - Discrete 5

Source:

Alice, Bob, and Chris each roll

4

dice. Each only knows the result of their own roll. Alice claims that there are at least

5

3

among the dice rolled. Bob has

1

six and no threes, and knows that Alice wouldn’t claim such a thing unless he had at least

2

3

. Bob can call Alice a liar, or claim that there are at least

6

3

, but Chris says that he will immediately call Bob a liar if he makes this claim. Bob wins if he calls Alice a liar and there aren't at least

5

3

, or if he claims there are at least

6

3

, and there are. What is the probability that Bob loses no matter what he does?

combinatoricsprobability

BMT 2014 Spring - Individual 5

Source:

Fred and George are playing a game, in which Fred flips

2014

coins and George flips

2015

coins. Fred wins if he flips at least as many heads as George does, and George wins if he flips more heads than Fred does. Determine the probability that Fred wins.