Problem

Source: China TST 2004 Quiz

Tags: inequalities, combinatorics unsolved, combinatorics



Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k - set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i + A) \cap (x_j + A) = \emptyset$, where $ x + A = \{ x + a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ - set}$($ i = 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t = \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} + \frac {1}{k_2} + \cdots + \frac {1}{k_t} \geq 1$.