Question 43
Prove that if $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \geq 0$ and $c_{k}=\prod_{i=1}^{k} b_{i}^{\frac{1}{k}}$, $1 \leq k \leq n$, then
$$
n c_{n}+\sum_{k=1}^{n} k\left(a_{k}-1\right) c_{k} \leq \sum_{k=1}^{n} a_{k}^{k} b_{k}
$$
$$
n c_{n}+\sum_{k=1}^{n} k\left(a_{k}-1\right) c_{k} \leq \sum_{k=1}^{n} a_{k}^{k} b_{k}
$$