Brazil

We have $$5(x_1^{2005}+x_2^{2005}+x_3^{2005}+x_4^{2005}+x_5^{2005})\ge (x_1^5+x_2^5+x_3^5+x_4^5+x_5^5)(x_1^{2000}+x_2^{2000}+x_3^{2000}+x_4^{2000}+x_5^{2000})$$ by Chebyshev. Also, $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5\ge 5x_1{x}_2{x}_3{x}_4{x}_5$$ by AM-GM and $$\frac{x_1^{2000}+x_2^{2000}+x_3^{2000}+x_4^{2000}+x_5^{2000}}{5}\ge {\left(\frac{x_1^{125}+x_2^{125}+x_3^{125}+x_4^{125}+x_5^{125}}{5}\right)}^{16}$$ Combining these inequalities gives $C\le 5^{15}$. But substituting $x_1=x_2=x_3=x_4=x_5=1$ gives $C\ge 5^{15}$. Thus $C=5^{15}$.