Question 2
Let $k$ be a positive integer. Define $(x_n)_{n\ge1}$ by $x_1=1$, and, for each $n\ge1$, let $x_{n+1}$ be the least positive integer not in $\{x_1;x_2;x_3;\ldots;x_n;x_1+k;x_2+2k;x_3+3k;x_n+nk\}$. Prove that there exists a real number $a$ such that $x_n=\lfloor na\rfloor$ for every positive integer $n$.
Editorial notation note: the printed list omits the continuation dots between its last two shifted terms. The source solution uses all terms $x_i+ik$ for $1\le i\le n$; that convention is used below.
Editorial notation note: the printed list omits the continuation dots between its last two shifted terms. The source solution uses all terms $x_i+ik$ for $1\le i\le n$; that convention is used below.