Question 35
Let $n \in \mathbb{N}^*$. Prove that
$$
\frac{a}{(n+1)a+b+c} + \frac{b}{a+(n+1)b+c} + \frac{c}{a+b+(n+1)c} \le \frac{1}{2}\sqrt{\frac{3}{n}},
$$
for every $a, b, c \in (0, \infty)$. Find the cases of equality.
$$
\frac{a}{(n+1)a+b+c} + \frac{b}{a+(n+1)b+c} + \frac{c}{a+b+(n+1)c} \le \frac{1}{2}\sqrt{\frac{3}{n}},
$$
for every $a, b, c \in (0, \infty)$. Find the cases of equality.