MathDB

2017 Team #8: floor(alpha^n) is divisible by 2017

Source:

2/19/2017

Does there exist an irrational number

\alpha > 1

such that

\lfloor \alpha^n \rfloor \equiv 0 \pmod{2017}

for all integers

n \ge 1

?

floor function

2017 Algebra/NT #8: (a+b)(a+b+1)/ab is an integer

Source:

2/19/2017

Consider all ordered pairs of integers

(a,b)

such that

1\le a\le b\le 100

and

\frac{(a+b)(a+b+1)}{ab}

is an integer.
Among these pairs, find the one with largest value of

b

. If multiple pairs have this maximal value of

b

, choose the one with largest

a

. For example choose

(3,85)

over

(2,85)

over

(4,84)

. Note that your answer should be an ordered pair.

2017 Geometry #8: Maximize sin A/2

Source:

2/19/2017

Let

ABC

be a triangle with circumradius

R=17

and inradius

r=7

. Find the maximum possible value of

\sin \frac{A}{2}

.

geometrycircumcircleinradius

2017 Theme #8

Source:

5/8/2018

Undecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

3D geometrygeometry

2017 Combinatorics #8: Frogs Making Friends with Each Other

Source:

2/20/2017

Kelvin and

15

other frogs are in a meeting, for a total of

16

frogs. During the meeting, each pair of distinct frogs becomes friends with probability

\frac{1}{2}

. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the

16

frogs, the number of friends they made during the meeting is a multiple of

4

. Say that the probability of the situation being cool can be expressed in the form

\frac{a}{b}

, where

a

and

b

are relatively prime. Find

a

.

probabilitynumber theoryrelatively prime

2017 General #8

Source:

5/8/2018

Marisa has a collection of

2^8-1=255

distinct nonempty subsets of

\{1, 2, 3, 4, 5, 6, 7, 8\}

. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process

2^8-2=254

times until there is only one set left in the collection. What is the expected size of this set?

combinatorics

2017 Guts #8: Single-elimination tournament

Source:

2/21/2017

You have

128

teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the

128

teams in the tournament is equally strong, so during each match, each team has an equal probability of winning.
Now, the

128

teams are randomly put into the bracket.
What is the probability that the Engineers play the Crimson sometime during the tournament?

combinatorics

8

Problems(7)