IMC 2006 Problem 5

Let $a, b, c, d$ three strictly positive real numbers such that $$a^{2}+b^{2}+c^{2}=d^{2}+e^{2},$$ $$a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.$$ Compare $$a^{3}+b^{3}+c^{3}$$ with $$d^{3}+e^{3},$$