Question 134
For $n \ge 3$, a *special n-triangle* is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special 3-triangle:
$$
\begin{array}{c c c}
& 41 & \\
23 & 19 & \\
43 & 17 & 47
\end{array}
$$
Note that a special $n$-triangle contains $3(n-1)$ numbers.
An infinite set $A$ of positive integers is a *special set* if, for each $n \ge 3$, the smallest $3(n-1)$ numbers of $A$ can be used to form a special $n$-triangle.
Show that the set of positive integers that are not multiples of 2023 is a special set.
$$
\begin{array}{c c c}
& 41 & \\
23 & 19 & \\
43 & 17 & 47
\end{array}
$$
Note that a special $n$-triangle contains $3(n-1)$ numbers.
An infinite set $A$ of positive integers is a *special set* if, for each $n \ge 3$, the smallest $3(n-1)$ numbers of $A$ can be used to form a special $n$-triangle.
Show that the set of positive integers that are not multiples of 2023 is a special set.