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China TST 2003 inequality

Source: China Team Selection Test 2003, Day 1, Problem 3

October 13, 2005
inequalitiesvectorinductionfunctionceiling functioninequalities unsolved

Problem Statement

Suppose A⊂{(a1,a2,…,an)∣ai∈R,i=1,2…,n}A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}. For any α=(a1,a2,…,an)∈A\alpha=(a_1,a_2,\dots,a_n)\in A and β=(b1,b2,…,bn)∈A\beta=(b_1,b_2,\dots,b_n)\in A, we define γ(α,β)=(∣a1−b1∣,∣a2−b2∣,…,∣an−bn∣), \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), D(A)={γ(α,β)∣α,β∈A}. D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. Please show that ∣D(A)∣≥∣A∣|D(A)|\geq |A|.
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