Question 89
Prove that for any integer $n \gt 1$ there exists such a sequence of positive integers $a_1 \le a_2 \le \dots \le a_m = n$, $m \gt 1$ such that
$$
5(a_1^2 + a_2^2 + \dots + a_m^2) - 4(a_1a_2 + a_2a_3 + \dots + a_{m-1}a_m) \le 4n^2 + \frac{1}{2}(m+1).
$$
$$
5(a_1^2 + a_2^2 + \dots + a_m^2) - 4(a_1a_2 + a_2a_3 + \dots + a_{m-1}a_m) \le 4n^2 + \frac{1}{2}(m+1).
$$