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Represent as Differences

Source: 2021 China TST, Test 1, Day 1 P3

March 17, 2021
number theoryAdditive Number Theory

Problem Statement

Given positive integer nn. Prove that for any integers a1,a2,⋯ ,an,a_1,a_2,\cdots,a_n, at least ⌈n(n−6)19⌉\lceil \tfrac{n(n-6)}{19} \rceil numbers from the set {1,2,⋯ ,n(n−1)2}\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \} cannot be represented as ai−aj(1≤i,j≤n)a_i-a_j (1 \le i, j \le n).
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