MathDB
Sequence

Source: USAMO 1997

October 9, 2005
floor functioninductioninequalitiesalgorithmalgebra proposedalgebra

Problem Statement

Suppose the sequence of nonnegative integers a1,a2,…,a1997a_1, a_2, \ldots, a_{1997} satisfies ai+aj≤ai+j≤ai+aj+1 a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 for all i,j≥1i,j \geq 1 with i+j≤1997i + j \leq 1997. Show that there exists a real number xx such that an=⌊nx⌋a_n = \lfloor nx \rfloor (the greatest integer ≤nx\leq nx) for all 1≤n≤19971 \leq n \leq 1997.
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