KMO 2018 Problem 4

Find all real values of $K$ which satisfies the following.

Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$ .

(i). $0 < a_n < n^K$ . (ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$ .

Then, there exists a positive integer $N$ such that for all integers $n>N$ , $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$