| 1. Given a field F with operations + (identity 0) and x (identity 1). Express x using the binary operation - (minus) and the unary operation a-1 (multiplicative inverse). | |
| 2. Find a single operation from which +, -, x and / can all be derived. | |
| 3. If additions and subtractions are free, how many real multiplications are needed to multiply two complex numbers? | |
| 4. A bug either splits into two perfect copies of itself or dies. If the probability of splitting is p (and is independent of the bug's ancestry), what is the probability that a bug's descendants die out? | |
| 5. Given any n distinct points in the plane, show that one of the angles determined by them falls in the closed interval [0, π/n]. | |
| 6. Prove that every sequence of real numbers contains a monotone subsequence. | |
| 7. Define fn,1(x) = x + x2/n, fn,r+1 = fn,1( fn,r(x) ) for 1 ≤ r ≤ n. What is the limit of fn,n(x) as n tends to infinity? | |
| 8. Given any p in the closed interval [0, 1], devise an experiment using a fair coin which has success probability p. | |
| 9. Two players alternately choose a binary digit bi. The digits are used to construct a real number b = 0.b1b2b3... . The first player wins iff b is transcendental. Who wins? | |
| 10. A traffic light is green for 30 seconds, then red for 30 seconds, then green for 30 seconds and so on. What is your expected delay? |
Donald J Newman, A problem seminar (Springer, problem books in mathematics, 1982). Highly recommended. It comes with short hints, giving the key idea, and solutions which are concise but motivated (in other words, they often explain how you might have found the solution yourself).
John Scholes
jscholes@kalva.demon.co.uk
2 Jan 1999