| 41. C is a simple plane arc of length > 1. Prove that for some n > 1 we can find at least n + 1 points on C such that the distance between each pair is at least 1/n. | |
| 42. You are given 4 coins, each of which may weigh 10gm or 9gm. Given an accurate scale (which gives the weight of a group of coins), show how to determine the weight of each coin using only 3 weighings in total. | |
| 43. α and β are reals such that [α, β] contains no integers. Prove that we can find a positive integer n such that [nα, nβ] contains no integers and has |nα - nβ| ≥ 1/6. | |
| 44. Prove that the [(√2 + 1)], [(√2 + 1)2], [(√2 + 1)3], ... are alternately even and odd integers. | |
| 45. Let ak = k + [n/k]. Show that the smallest element of {a1, a2, ... , an} equals [√(4n + 1)]. | |
| 46. α and β are positive and irrational. Show that the sets {[α], [2α], [3α], ... } and {[β], [2β], [3β], ... } have null intersection and union {1, 2, 3, ... } iff 1/α + 1/β = 1. | |
| 47. Select a subset A = {a1, a2, a3, ... } of the positive integers as follows. Take a1 = 1. Then reject a1 + 1 = 2. Then take a2 to be the smallest remaining integer, 3, and reject a2 + 2 = 5. In general, after selecting an, reject an + n, and select an+1 to be the smallest remaining integer. The first few members of A are 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, ... . Find a formula for an. | |
| 48. A doublet is a positive integer all of whose prime factors occur to the second power or higher. Prove that there are infinitely many pairs of consecutive doublets. | |
| 49. Let N be the set of positive integers. If A and B are subsets of N, then we say they are almost disjoint iff A ∩ B is finite. How many subsets of N can we find every pair of which are almost disjoint? | |
| 50. n people each wish to have an equal share of a cake. A knife is available, but no measuring equipment. What procedure will satisfy each person that he has been treated fairly? For example, for n = 2, one person divides the cake into two pieces, and the other person chooses which piece to take. [Assume there are no complications with icing, shape etc. All that counts is the volume of the piece.] |
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).
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John Scholes
jscholes@kalva.demon.co.uk
25 Sep 1999