Let a,b,c and d be distinct real numbers such that
a+b+c+d=3
a
^{2}+b
^{2}+c
^{2}+d
^{2}=45
Find the value of the expression
a^{5} b^{5} c^{5} d^{5}
 +  +  + 
(ab)(ac)(ad) (ba)(bc)(bd) (ca)(cb)(cd) (da)(db)(dc)
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 20131009 11:39:22 