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Non-decreasing unbounded sequence of non-negative integers

Source: Balkan MO 1999, Problem 4

April 24, 2006
inductioninequalitiesinequalities proposed

Problem Statement

Let {an}n≥0\{a_n\}_{n\geq 0} be a non-decreasing, unbounded sequence of non-negative integers with a0=0a_0=0. Let the number of members of the sequence not exceeding nn be bnb_n. Prove that (a0+a1+⋯+am)(b0+b1+⋯+bn)≥(m+1)(n+1). (a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1).
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