The Art and Craft of Problem Solving

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 +X 5 = 12

Xl +X 2 +2x 3 +X 4 + X 5 = 24

Xl +X 2 +X 3 + 2x 4 +X 5 = 48

Xl +X 2 +X 3 +X 4 + 2x 5 = (^96).
Solution: The standard procedure for solving systems of equations by hand is to
substitute for and/or eliminate variables in a systematic (and tedious) way. But notice
that each equation is almost symmetric, and that the system is symmetric as a whole.
Just add together all five equations; this will serve to symmetrize all the coefficients:
(^6) (Xl +X 2 +X 3 +X 4 +X 5 ) = 6(1 + (^2) + 4 + 8 + 1 6),
so
Xl +X 2 +X 3 +X 4 +X 5 = (^31).
Now we can subtract this quantity from each of the original equations to immediately
get Xl = 6 -31,X 2 = 12 - 31, etc. _
You have seen the define a function tool in action in Example 3.3. 8 on page 89
and in Example 5.4.2 on page 167. Here is another example, one that also employs a
large dose of symmetry.
Example 5.2.19 Show, without m ultiplying out, that
b-c c-a a-b (a- b) (b-c)(a-c)
--+--+--=.
abc abc
Solution: Even though it is easy to mUltiply out, let us try to find a more elegant
approach. Notice how the right-hand side factors. We can deduce this factorization by
defining
b-c c -x x-b
f(x) := -
x

• 
  
  
  
  • 
    
    
    
    • b
      
      
      • 
        
        
        
        • 
          
          
          
          • c
            
            
            • '
            
            
            
            
          
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  
  

Notice that f(b) = f(c) = O. By the factor theorem, if we write f(x) as a quotient of

polynomials

f(x) =

P

b

(x)

,

X c

then P(x) must have X - b and X -c as factors. Also, it is clear that P(x) has degree 3.

Plugging in X = a into f(x) , we conclude that

b-c c - a a-b (a-b)(a -c)R(a)

• a
  
  
  • +-
    b
    
    
    • +-
      c
      
      
      • =
        abc '
        where R(x) is a linear polynomial. By symmetry, we could also define the function
        x-c c-a a-x
        g(x): =-+-+-,
        a X c