Question 7
Let $N$ be a positive integer. Prove that there exist three permutations $a_{1}, a_{2}, \ldots, a_{N}$; $b_{1}, b_{2}, \ldots, b_{N}$; and $c_{1}, c_{2}, \ldots, c_{N}$ of $1,2, \ldots, N$ such that
$$
\left|\sqrt{a_{k}}+\sqrt{b_{k}}+\sqrt{c_{k}}-2 \sqrt{N}\right|\lt 2023
$$
for every $k=1,2, \ldots, N$.
$$
\left|\sqrt{a_{k}}+\sqrt{b_{k}}+\sqrt{c_{k}}-2 \sqrt{N}\right|\lt 2023
$$
for every $k=1,2, \ldots, N$.