The Art and Craft of Problem Solving

144 CHAPTER 5 ALGEBRA We can define "subtraction" for sets in the following natural way: A-B:={aEA:a�B}; in other words, A - B i ...

5.1 SETS, NUMBERS, AND FUNCTIONS 145 closure. The natural numbers N are not closed under subtraction, Z is not closed under divi ...

146 CHAPTER 5 ALGEBRA logarithm 10gbY' For example, if b = 3, then log 3 81 = 4 because x = 4 is the unique^3 solution to 3x = 8 ...

(b) Between any two irrational numbers there is a rational number. 5.1.5 The number of elements of a set is called its cardinali ...

148 CHAPTER 5 ALGEBRA 32 = (X+y)^2 =�+2xy+l, and since.xy = 3, we have � + y^2 = 3. Consequently, (^3) ·3= (x+ y)(� +l) =.J +�y+ ...

5.2 ALGEBRAIC MANIPULATION REVISITED 149 Many problems involve combinations of these fonnulas, along with basic strate­ gies (fo ...

150 CHAPTER 5 ALGEBRA The tactic of extracting squares includes many tools in addition to completing the square. Here are a few ...

5.2 ALGEBRAIC MANIPULATION REVISITED 151 equation x^4 + x^3 + x^2 + x + 1 = 0 to two quadratic equations. Here are a few more ex ...

152 CHAPTER 5 ALGEBRA Example 5.2.18 (AIME 198 6) Solve the system of equations 2x 1 +X 2 +X 3 +X 4 +x 5 = 6 Xl + 2x 2 +X 3 +X 4 ...

5.2 ALGEBRAIC MANIPULATION REVISITED 153 and we have g(a) = g(e) = 0, yielding the factorization b-e e-a a-b ( b-a)( b-e)Q(a) -a ...

154 CHAPTER 5 ALGEBRA Example 5.2.21 (AIME 198 6) The polynomial 1 -x+x^2 -..J + ... +xI^6 _xI^7 may be written in the form ao + ...

5.2 ALGEBRAIC MANIPULATION REVISITED 155 These tenns look suspiciously like they came from squares of binomials. For example, (X ...

156 CHAPTER 5 ALGEBRA Problems and Exercises 5.2.24 (AIME 1987) Find 3.?l if x, y are integers such thatl+3x^2 l = 30x^2 +517. 5 ...

5.3 SUMS AND PRODUCTS 157 • � d^2 = 12 + (^22) + (^52) + 102 , since dllO underneath the summation symbol means lito "d ranges t ...

158 CHAPTER 5 ALGEBRA Upon adding, we immediate deduce that the intuitively reasonable fact that the sum is equal to the average ...

5.3 SUMS AND PRODUCTS 159 The entire sum is ( 1 �) + (� �) + (� _ �) + ... + (� __ 1 ) 223 34 99 100 ' 1 and all terms cancel ...

160 CHAPTER 5 ALGEBRA n n and we can solve for}; p. We still need to sum the arithmetic series }; (3 j + I), but J=! J=! we alre ...

5.3 SUMS AND PRODUCTS 161 valid if and only if Irl < I. This is a simple consequence of the formula for a finite geometric se ...

162 CHAPTER 5 ALGEBRA an expression, with whatever method works (adding zero, multiplying by one, adding or subtracting a bit, e ...

5.3 SUMS AND PRODUCTS 163 When we are concerned about convergence, the first few terms do not matter at all. In fact, the first ...