Problem

Source: CWMO 2003, Problem 2

Tags: vector, inequalities



Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i = 1}^{2n - 1} (a_{i + 1} - a_i)^2 = 1$. Find the greatest possible value of $ (a_{n + 1} + a_{n + 2} + \ldots + a_{2n}) - (a_1 + a_2 + \ldots + a_n)$.