The Art and Craft of Problem Solving

124 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS (h) Observe that if a E C and r E �, the set of points a + reit, where 0 ::; t : ...

4.2 COMPLEX NUMBERS 125 Let AB be a diameter oj a circle with diameter k. Consider a right triangle ABC with right angle at B. L ...

126 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS In other words, if you start with z and then reciprocate, negate, translate by 1 ...

4.2 COMPLEX NUMBERS 127 4.2. 11 For each positive integer n, there are n different nth roots of unity, namely 1 , S', S'^2 , S'^ ...

128 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS Solution: Our solution illustrates a useful application of viewing complex num­ ...

is the annulus v-u:::; Izl:::; v+u 4.2 COMPLEX NUMBERS 129 centered at the origin, i.e. the set of points whose distance to the ...

130 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS Then there is a triangle with side lengths a, b, c. 2. Let 00 = e^2 7r:i/^3. Wit ...

4.2 COMPLEX NUMBERS 131 have the same state, then we can change the state of these n/ d lights. The sum of these is sa + sa+d + ...

132 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS change interpretation. Recall that mUltiplica­ tion by i means "rotate by 90 deg ...

4.3 GENERATING FUNCTIONS 133 • "Local" knowledge about the coefficients of a polynomial or power series f(x) often provides "glo ...

134 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS This identity can of course be proven in other ways (and you should know some of ...

4.3 GENERATING FUNCTIONS 135 This is the generating function of the original sequence, but shifted. In other words, the coeffici ...

136 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS Partitions Consider the following polynomial product: P(x) := (2 + 3x+ 1)(^2 + 2 ...

4.3 GENERATING FUNCTIONS 137 We claim that A(x)B(x) is the generating function for the sequence Uo, Ul, U 2 ,' ... This isn't ha ...

138 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS Example 4.3.6 Let n be any positive integer. Show that the set of weights 1,3, 3 ...

4.3 GENERATING FUNCTIONS 139 When multiplied out, each of the resulting 3. 3. 3 = 27 terms will be of the form x!1 , where a is ...

140 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS Now it is clear that the weights 1,3,3^2 , ... , 3r can be used to weigh any int ...

On the other hand, we can write 4.3 GENERATING FUNCTIONS 141 U(x) = (1 +x) (1 +x^2 )(1 +x^3 )(1 +x^4 )(^1 +x^5 ) ... = (\�:) C = ...

142 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS In/ 3 J ( ) (c) Find a simple fonnula for the sum � ;.. }=o } (d) Generalize! 4. ...

Chapter 5 Algebra You probably consider yourself an old hand at algebra. Nevertheless, you may have picked up some bad habits or ...