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)

The equations don't give enough info to actually find the values of the variables.  If we are told the expression has a definite value then all we need are values of the variables that fit the equations.

So lets let c=0 and d=0. 
a+b=3
a^2+b^2=45
One solution is a=6, b=-3

The expression becomes
6^5/(9*6*6)+(-3)^5/(-9*-3*-3) = 27

[If this value depends on my choice of c and d, then I haven't really answered the question.  But I suspect it doesn't.]

Posted by Jer on 2013-10-09 11:39:22