9

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #9

Source:

A sequence of positive integers is defined by

a_0=1

a_{n+1}=a_n^2+1

n\ge0

\text{gcd}(a_{999},a_{2004})

number theorygreatest common divisormodular arithmetic

2004 Calculus #9

Source:

Find the positive constant

c_0

such that the series

\displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n}

c>c_0

and diverges for

0<c<c_0

calculuslimitratioalgebrapolynomiallogarithmsabsolute value

2004 Combinatorics #9

Source:

A classroom consists of a

5\times5

array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?

2004 HMMT Geometry # 9

Source:

Given is a regular tetrahedron of volume

1

. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

2004 General, part 1 #9

Source:

4

white balls and

2

red balls. Urn B contains

3

3

black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

2004 HMMT General, part 2 #9 reflecting regular tetrahedron through center

Source:

Given is a regular tetrahedron of volume

1

. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

geometry3D geometryVolumetetrahedron