Math 4802
Fall 2007
Combinatorics Problems (due Tuesday, October 30) 1. Dene a selsh set to be a set which has its own cardinality (number of elements) as an element. A selsh set is called minimal if none of its proper subsets is selsh. Find the number of
Math 4802
Fall 2007
Infinite Series Problems (due Tuesday, October 23) 1. Evaluate the innite sum
n=1
n2 . 2n
2. Suppose that a sequence a1 , a2 , a3 , . . . of positive real numbers satises an a2n + a2n+1 for all n 1. Prove that the series an diverges. n
Math 4802
Fall 2007
Probability Problems Due Tuesday, September 11: 1. Shue an ordinary deck of 52 playing cards, and then turn the cards face up one at a time. On average, how many cards are required to produce the rst ace? 2. Show that the probability t
Math 4802
Fall 2007
Polynomial Problems (due Tuesday, October 16) 1. Determine all polynomials P (x) with complex coecients such that P (0) = 0 and P (x2 + 1) = (P (x)2 + 1. 2. Suppose the polynomial f (x) = ax3 + bx2 + cx + d has integer coecients a, b,
Math 4802
Fall 2006
Pigeonhole Principle Problems In-class Problems: 1. Suppose that 17 playing cards are chosen at random from a standard deck of 52 cards. Show that these cards must contain a ush. 2. Five darts are thrown onto a square dartboard having
Math 4802
Fall 2007
Number Theory Problems 1. a. Find all positive integers n for which 7 divides 2n 1. b. Show that there is no positive integer n for which 7 divides 2n + 1. 2. Find all prime numbers p whose base 10 expansion consists of alternating 1s
Math 4802
Fall 2007
Inequalities Problems (due Tuesday, November 6) 1. If a, b, c are positive real numbers, prove that a4 + b4 + c4 abc(a + b + c) . 2. Given that a, b, c, d, e are real numbers for which a+b+c+d+e =8 2 2 2 2 2 a + b + c + d + e = 16 , nd