Question 87
Prove that there exists constant $D \gt 0$ satisfying the following: for any positive integer $m$ and $N = \frac{m(m+1)}{2}$, there exist three permutations $a_1, a_2, \dots, a_N$; $b_1, b_2, \dots, b_N$; and $c_1, c_2, \dots, c_N$ of $1, 2, \dots, N$ such that
$$
\left| \sqrt{a_k} + \sqrt{b_k} + \sqrt{c_k} - 2\sqrt{N} \right| \lt D
$$
for every $k = 1, 2, \dots, N$.
$$
\left| \sqrt{a_k} + \sqrt{b_k} + \sqrt{c_k} - 2\sqrt{N} \right| \lt D
$$
for every $k = 1, 2, \dots, N$.