MathDB

2014 HMMT #2: Closest integer

Source:

2/23/2014

Find the integer closest to

\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}

HMMT

2014 Combinatorics #2: Number of Committees

Source:

2/23/2014

There are

10

people who want to choose a committee of 5 people among them. They do this by first electing a set of

1, 2, 3,

or

4

committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assuming that people are distinguishable? (Two committees that have the same members but different sets of leaders are considered to be distinct.)

countingdistinguishability

2014 Geometry #2: Point P and line l

Source:

2/25/2014

Point

P

and line

\ell

are such that the distance from

P

to

\ell

is

12

. Given that

T

is a point on

\ell

such that

PT = 13

, find the radius of the circle passing through

P

and tangent to

\ell

at

T

.

geometryanalytic geometrypower of a point

2014 Guts #2: expected multiples of 6

Source:

2/25/2014

[4] Let

x_1,x_2,\ldots,x_{100}

be defined so that for each

i

,

x_i

is a (uniformly) random integer between

1

and

6

inclusive. Find the expected number of integers in the set

\{x_1,x_1+x_2,\ldots,x_1+x_2+\cdots+x_{100}\}

that are multiples of

6

.

probabilityexpected value

2014 Team #2: Infinite Sequence of Integers

Source:

3/2/2014

Let

a_1,a_2,\ldots

be an infinite sequence of integers such that

a_i

divides

a_{i+1}

for all

i\geq 1

, and let

b_i

be the remainder when

a_i

is divided by

210

. What is the maximal number of distinct terms in the sequence

b_1,b_2,\ldots

?

2

Problems(5)