This has been recently posted on aops:
(China Southeast Mathematical Olympiad 2018 Grade 11 P8) Given a positive real $C \geq 1$ and a sequence $a_1, a_2, ...$ of nonnegative real numbers satisfy $$\left|x\ln{x}-\sum_{k=1}^{[x]} \left[\frac{x}{k}\right]a_k\right| \leq Cx,$$ where $[x]$ is the floor function of $x.$ Prove that for any real $y \geq1,$ $$\sum_{k=1}^{[y]} a_k<3Cy.$$ An obvious step might be this well-known identity: $$\sum_{k=1}^{[x]} \left[\frac{x}{k}\right]a_k=\sum_{k=1}^{[x]} F\left(\frac{x}{k}\right),$$ where $F(x)$ is the sum of all $a_k, k\leq x.$ Because this is olympiad-level, there should be an elementary solution but perhaps the bound could be improved using higher mathematics?