MathDB

Let

k

be a positive integer. Set

A \subseteq \mathbb{Z}

is called a k \minus{ set} if there exists

x_1, x_2, \cdots, x_k \in \mathbb{Z}

such that for any

i \neq j

, (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset, where x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}. Prove that if

A_i

is k_i \minus{ set}( i \equal{} 1,2, \cdots, t), and A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}, then \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1.

Problem Statement