1: Suppose x + y = 1 and xy = -1
(a) Find x2 + y2.
(b) Find x3 + y3.
(c) Find x10 + y10.
2: (a) Let S be the set of all numbers in the sequence 1, 2, 3, ... 100 that have no 0 digits. Let T be the set of all numbers in the sequence 1, 2, 3, ..., 1000 that have exactly one digit that is 0. Show that the total number of digits in S is the same as the size of T.
(b) Show that for every positive integer n, the total number of digits in the sequence 1, 2, 3, ... , 10n is equal to the total number of zero digits in the sequence 1, 2, 3, .., 10n+1.
3: Let n be a fixed positive integer. How many ways are there to write n as a sum of positive integers, n = a1 + a2 + · · · + ak, with k an arbitrary positive integer and a1 ≤ a2 ≤ · · · ≤ ak ≤ a1 + 1. For example, if n = 4, there are four ways: 4, 2+2, 1+1+2, 1+1+1+1.
4: Greedy Pirates: You have 1000 pirates, who are all extremely greedy, heartless, and perfectly rational. They're also aware that all the other pirates share these characteristics.
They're all ranked by the order in which they joined the group, from pirate one down to pirate one thousand. They've stumbled across a huge horde of treasure, and they have to decide how to split it up. Every day they will vote to either kill the lowest ranking pirate, or split the treasure up evenly among the surviving pirates. If 50% or more of them vote to split it, the treasure gets split. Otherwise, they kill the lowest ranking pirate and repeat the process until half or more of the pirates decide to split the treasure.
The question, of course, is at what point will the treasure be split, and what will the precise vote be?
5: Dissect a square into n isosceles right triangles of different sizes. How small can n be?