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every positive integer k can be written as x_j -x_i for some i, j if x_{n+1}<=2n

Source: Austrian Polish 1980 APMC

April 26, 2020
Sequenceinequalitiesnumber theory

Problem Statement

A sequence of integers 1=x1<x2<x3<...1 = x_1 < x_2 < x_3 <... satisfies xn+1≤2nx_{n+1} \le 2n for all nn. Show that every positive integer kk can be written as xj−xix_j -x_i for some i,ji, j.
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