Suponga que $a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+$ tal que $a_i\geq a_{101-i}\,(i=1,2,\cdots,50).$ Sea $x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).$ Demuestra que $$x_1x_2^2\cdots x_{99}^{99}\leq 1.$$