66th Belarusian Mathematical Olympiad

The sequence $a_1,a_2,\dots$ of positive real numbers satisfies $$a_{k+1}\ge \frac{k{a}_k}{a_k^2+k-1}$$ for every positive integer $k$. Prove that $a_1+a_2+\cdots +a_n\ge n$ for every $n\ge 2$.