Quiz for 2008 Brainstorming and Barnstorming Applicants
1. Prove that the number of way of writing a positive interger n as a ordered sum of 1's and 2's is the same as the number of ways of writing n+2 as an ordered sum of integers greater than 1.
2. The real-coefficient polynomial p(x) has the property that if q(x) is any real-coefficient polynomial, p(q(x)) = q(p(x)) (another way of saying this is the p(x) is in the center of the monoid of real-coefficient polynomials under composition). Find all p(x) with this property.
3. Suppose a(n) is a sequence of positive integers. Show that the lim sup of the sequence
is greater than or equal to e.
4. Choose either A or B.
A. In the following approximate the density of both liquid water and of ice by 1.0 grams per cubic cm, and assume that the fine glass container in which the water is held is made of a glass with a density of 3.0 grams per cubic cm.
- If the glass is 10. cm tall, has a base 1.0 cm thick, an outside diameter of 3.0 cm, and walls 1.0 mm thick is half full of ice, find the tipping point: angle at which if the glass, resting on a horizonatal table, is tilted, the center of mass is exactly above the point at which the edge touches the table.
- Answer the same question if the glass is half full of liquid water.