5

Part of 2008 Harvard-MIT Mathematics Tournament

Problems(5)

Source:

Let f(x) \equal{} x^3 \plus{} x \plus{} 1. Suppose

g

is a cubic polynomial such that g(0) \equal{} \minus{} 1, and the roots of

g

are the squares of the roots of

f

g(9)

algebrapolynomialVietafunction

2008th Derivative of Trig

Source: HMMT 2008 Calculus Problem 5

(4) Let f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right) for all real numbers

x

f^{(2008)}(0)

f

2008

times and then evaluated at x \equal{} 0).

calculusderivativetrigonometrycalculus computations

Subset of Integers

Source:

S

be the smallest subset of the integers with the property that

0\in S

x\in S

3x\in S

and 3x \plus{} 1\in S. Determine the number of non-negative integers in

S

2008

Folded Paper Cutting

Source:

A piece of paper is folded in half. A second fold is made at an angle

\phi

0^\circ < \phi < 90^\circ

) to the first, and a cut is made as shown below. 12881 When the piece of paper is unfolded, the resulting hole is a polygon. Let

O

be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at

O

, and also that \angle XOY \equal{} 144^\circ, where

X

Y

are the the vertices of the hole adjacent to

O

. Find the value(s) of

\phi

Moderator + Vandal

Source:

A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a

2/3

chance of catching each individual error still in the article. After

3

days, what is the probability that the article is error-free?

articlesprobability