Question 120
The positive integer $k$ and the set $A$ of different integers from $1$ to $3k$ inclusive are such that there are no distinct $a, b, c$ in $A$ satisfying $2b = a + c$. The numbers from $A$ in the interval $[1, k]$ will be called small; those in $[k+1, 2k]$ - medium and those in $[2k+1, 3k]$ - large. Is it always true that there are no positive integers $x$ and $d$ such that if $x, x+d$ and $x+2d$ are divided by $3k$ then the remainders belong to $A$ and those of $x$ and $x+d$ are different and are:
a) small?
b) medium?
c) large?
(In this problem we assume that if a multiple of $3k$ is divided by $3k$ then the remainder is $3k$ rather than $0$.)
a) small?
b) medium?
c) large?
(In this problem we assume that if a multiple of $3k$ is divided by $3k$ then the remainder is $3k$ rather than $0$.)