Let $$a, b, c$$ be non zero real numbers such that $$a+b+c = 0$$ and $$a^3+b^3+c^3 = a^5+b^5+c^5$$. The value of $$\frac{5}{a^2+b^2+c^2}$$ is

With $$e_1 = a+b+c = 0$$, Newton's identities give $$p_2 = -2e_2$$, $$p_3 = 3e_3$$ and $$p_5 = -5e_2e_3$$. The condition $$p_3 = p_5$$ with $$e_3 = abc \neq 0$$ forces $$3 = -5e_2$$, so $$e_2 = -\frac{3}{5}$$ and $$a^2+b^2+c^2 = -2e_2 = \frac{6}{5}$$. Hence $$\frac{5}{a^2+b^2+c^2} = \frac{25}{6}$$, which is approximately 4.1667.