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Special n-triangles

Source: Irish Mathematical Olympiad 2023 Problem 2

May 15, 2023
combinatorics

Problem Statement

For n≥3n \geq 3, a special n-triangle is a triangle with nn distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because 41+23+43=43+17+47=47+19+4141 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41, the following is a special 33-triangle:
4141 23     1923\text{ }\text{ }\text{ }\text{ }\text{ }19 43     17     4743\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47
Note that a special nn-triangle contains 3(n−1)3(n - 1) numbers.
An infinite set AA of positive integers is a special set if, for each n≥3n \geq 3, the smallest 3(n−1)3(n - 1) numbers of AA can be used to form a special nn-triangle.
Show that the set of positive integers that are not multiples of 20232023 is a special set.
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