Problem

Source: China TST 1986, problem 2

Tags: inequalities, function, vector, algebra unsolved, algebra



Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: i) For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k = 1} a_k \cdot x_k \leq \sum^{n}_{k = 1} b_k \cdot x_k,$ ii) We have $ \sum^{s}_{k = 1} a_k \leq \sum^{s}_{k = 1} b_k$ for every $ s\in\left\{1,2,...,n-1\right\}$ and $ \sum^{n}_{k = 1} a_k = \sum^{n}_{k = 1} b_k$.