KöMaL 2012 A566

($a$) Prove that if $n \ge$2 and the product of the positive real numbers $a_2,a_3,\ldots,a_n$ is 1 then ${(1+a_2)}^2{(1+a_3)}^3 \cdots {(1+a_n)}^n > \frac{n^n{(n-1)}^{n-1}}{4^{n-1}}$. ($b$) Show an example for an integer $n \ge$2 and positive real numbers $a_2,a_3,\ldots,a_n$ having product 1 that satisfy ${(1+a_2)}^2{(1+a_3)}^3\ldots{(1+a_n)}^n < 1.000001 \cdot \frac{n^n{(n-1)}^{n-1}}{4^{n-1}}$.