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Moldova MO 2006, 9.1 (tricky ineq.)

Source: Moldova national olympiad 2006, grade 9, pr.1

September 21, 2016

inequalities

Problem Statement

Let

a,b,c

be positive real numbers such that

a+b+c=2005

. Find the minimum value of the expression:

E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}