Question 1
Determine the smallest real number $C$ such that the inequality
$$
C(x_1^{2005} + x_2^{2005} + x_3^{2005} + x_4^{2005} + x_5^{2005}) \geq x_1 x_2 x_3 x_4 x_5 (x_1^{125} + x_2^{125} + x_3^{125} + x_4^{125} + x_5^{125})^{16}
$$
holds for all positive real numbers $x_1, x_2, x_3, x_4, x_5$.
$$
C(x_1^{2005} + x_2^{2005} + x_3^{2005} + x_4^{2005} + x_5^{2005}) \geq x_1 x_2 x_3 x_4 x_5 (x_1^{125} + x_2^{125} + x_3^{125} + x_4^{125} + x_5^{125})^{16}
$$
holds for all positive real numbers $x_1, x_2, x_3, x_4, x_5$.