Question 97
Let $k\geqslant1$ be an integer, and let $A\subseteq\{1,2,\ldots,3k\}$ have the following property: for any $a,b,c\in A$, if $a+b=2c$, then $a=b=c$. For each integer $n\geqslant1$, let $r_k(n)$ be the smallest positive integer such that $3k$ divides $n-r_k(n)$. Call $n$ small if $n\leqslant k$, medium if $k+1\leqslant n\leqslant2k$, and large if $2k+1\leqslant n$. Suppose positive integers $x,d$ satisfy $r_k(x),r_k(x+d),r_k(x+2d)\in A$ and $r_k(x)\ne r_k(x+d)$.
Does the information above necessarily imply each of the following?
a) At least one of $r_k(x)$ and $r_k(x+d)$ is medium or large.
b) At least one of $r_k(x)$ and $r_k(x+d)$ is small or large.
c) At least one of $r_k(x)$ and $r_k(x+d)$ is small or medium.
Does the information above necessarily imply each of the following?
a) At least one of $r_k(x)$ and $r_k(x+d)$ is medium or large.
b) At least one of $r_k(x)$ and $r_k(x+d)$ is small or large.
c) At least one of $r_k(x)$ and $r_k(x+d)$ is small or medium.