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Find the value of ...... (Posted on 2006-05-14) Difficulty: 3 of 5
If a + b + c = 0, then find the value of

[(bc)/a + (ca)/b + (ab)/c].[a/(bc) + b/(ca) + c/(ab)]

No Solution Yet Submitted by Ravi Raja    
Rating: 4.0000 (1 votes)

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Hints/Tips The Process | Comment 1 of 6

Some hints for this puzzle are provided  in terms of  the comments furnished below.<o:p></o:p>

  1. At the outset, we observe that; a ,b and c cannot be all equal for obvious reasons ( tip: observe the denominators). Each of a, b and c must be non-zero. ( for same reason). Since, the problem asks for a definite solution, it follows that a,b and c cannot be all equal and each of  a,b and c must be non-zero. <o:p></o:p>
  2. Let us denote the given expression as S*T; where S = (b c)/a + (c a)/b + (a b)/c; and T = a/(b c) + b/(c a) + c/(a b)<o:p></o:p>
  3. Expand S and factorize its numerator, the denominator being a*b*c.<o:p></o:p>
  4.  Substitute p = a-b; q = b-c and r = c-a. Observe how each of a, b ,c  comes out in terms of p, q and r in view of  a+b+c = 0. Now, expand T in terms of p, q and r. Factorize numerator of  T ( in terms of p, q and r).<o:p></o:p>
  5. Express T in terms of a,b and c.<o:p></o:p>
  6. Finally evaluate S*T  to  obtain the desired result, which is a one-digit positive integer.<o:p></o:p>

<o:p> </o:p>

  Posted by K Sengupta on 2006-05-14 11:30:57
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