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Quintic Expression (Posted on 2013-10-09) Difficulty: 2 of 5
Let a,b,c and d be distinct real numbers such that

a+b+c+d=3
a2+b2+c2+d2=45

Find the value of the expression
       a5                b5                c5                d5
--------------- + --------------- + --------------- + --------------
(a-b)(a-c)(a-d)   (b-a)(b-c)(b-d)   (c-a)(c-b)(c-d)   (d-a)(d-b)(d-c)

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 7 of 8 |
Starting with the identity

a5(b-c)(b-d)(c-d) - b5(a-c)(a-d)(c-d) + c5(a-b)(a-d)(b-d) - d5(a-b)(a-c)(b-c)

= (a-b)(a-c)(a-d)(b-c)(b-d)(c-d)[a2+b2+c2+d2+ab+ac+ad+bc+bd+cd]

and then dividing by (a-b)(a-c)(a-d)(b-c)(b-d)(c-d), for distinct a,b,c,d:

a5/(a-b)(a-c)(a-d)+b5/(b-a)(b-c)(b-d)+c5/(c-a)(c-b)(c-d)+d5/(d-a)(d-b)(d-c)

            = a2 + b2 + c2 + d2 + ab + ac + ad + bc + bd + cd

            =  a2 + b2 + c2 + d2 + [(a + b + c + d)2 (a2 + b2 + c2 + d2)]/2

            = [a2 + b2 + c2 + d2 + (a + b + c + d)2]/2

            = [45 + 32]/2     using the given equalities

            = 27

.. with a little help from Mathematica in confirming the initial identity.

Why the restriction to real numbers?



  Posted by Harry on 2013-10-12 13:26:36
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