Question 52
Let $n \in \mathbb{N}$, $n \ge 2$, and let $a_1, a_2, \ldots, a_n$, and $b_1, b_2, \ldots, b_n$ be positive real numbers such that $a_1 + a_2 + \ldots + a_n = b_1 + b_2 + \ldots + b_n = S$.
a) Prove that $\sum_{k=1}^{n} \frac{a_k^2}{a_k + b_k} \ge \frac{S}{2}$.
b) Prove that $\sum_{k=1}^{n} \frac{a_k^2}{a_k + b_k} = \sum_{k=1}^{n} \frac{b_k^2}{a_k + b_k}$.
a) Prove that $\sum_{k=1}^{n} \frac{a_k^2}{a_k + b_k} \ge \frac{S}{2}$.
b) Prove that $\sum_{k=1}^{n} \frac{a_k^2}{a_k + b_k} = \sum_{k=1}^{n} \frac{b_k^2}{a_k + b_k}$.