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1000 consecutive integers among some sums

Source: Argentine National Olympiad 2014, Level 3, Problem 6

November 6, 2015
number theory

Problem Statement

Determine whether there exists positive integers a1<a2<⋅⋅⋅<aka_{1}<a_{2}< \cdot \cdot \cdot <a_{k} such that all sums ai+aj a_{i}+a_{j}, where 1 ≤i<j≤k\leq i < j \leq k, are unique, and among those sums, there are 10001000 consecutive integers.
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