The Art and Craft of Problem Solving

64 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS The small square obviously has half the area of the larger! - The simplest geometric s ...

3.1 SYMMETRY 65 Example 3.1. 6 Four bugs are situated at each vertex of a unit square. Suddenly, each bug begins to chase its co ...

66 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS Thus Consequently, the value that we seek is The next problem, from the 1995 IMO, ...

c C' F' 3.1 SYMMETRY 67 F A tactic that often works for geometric inequalities is to look for a way to compare the sum of severa ...

68 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS While the other students slowly added the numbers, little Carl disc overed a short­ cu ...

3.1 SYMMETRY 69 multiplicative inverse modulo p; i.e., if x is not a mUltiple of p then there is a unique y E {I, 2, 3, ... ,p - ...

70 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS so (1) becomes u^2 - 2 + u + 1 = 0, or u^2 + u - 1 = 0, which has solutions -1±V5 U= 2 ...

3.1 SYMMETRY 71 It is worth exploring the concept of cyclic permutation in more detail. Given an n-variable expression f(x\ ,X 2 ...

72 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS 3.1.18 A polynomial in several variables is called symmetric if it is unchanged when t ...

launched, without hitting the ceiling. This is possi­ ble because the projectile does not travel along straight lines, but inste ...

74 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS different lines and other shapes. A good problem solver always tries to organize this ...

3.2 THE EXTREME PRINCIPLE 75 less than 1. Show that all of these points lie within the interior or on the boundary of a triangle ...

76 CHAPTER (^3) TACTICS FOR SOLVING PROBLEMS Po It is interesting to look at the two people with the extreme handshake numbe ...

partner. 3.2 THE EXTREME PRINCIPLE 77 Now, let us remove X and Y from the party. If we no longer count handshakes involving thes ...

78 CHAPTER^3 TACTICS FOR SOLVING PROBLEMS Eventually, the sequence will fo rm a chain where each element will di­ vide the next ...

3.2 THE EXTREME PRINCIPLE 79 Now assume the inductive hypothesis that all sequences of length n will eventually become unchangin ...

80 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS Solution: There are several things to which we can apply extreme arguments. Since f(x) ...

3.2 THE EXTREME PRINCIPLE 81 is only one, (5, 93), and sure enough, their sum is an element of the sequence. So our sequence is ...

82 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS is less than or equal to n. Let's pick al + a6. If this sum is less than or equal to n ...

(b) Prove that it is impossible to find a tiling of the plane with infinitely many unit squares and finitely many (and at least ...