Let $x_1,\dots,x_n$ and $y_1,\dots,y_n$ be two increasing sequences of nonnegative real numbers with $x_i\leq y_i$ for all $i$. Is there a constant $c>0$ (independent of $n$) for which there exists some $r\geq 0$ (possibly dependent on $n$ and the sequences) such that $$\sum_{i: x_i\leq r\leq y_i}y_i+\sum_{i:x_i\geq r} x_i\geq c\sum_{i=1}^n y_i?$$
This is a discrete version of this question. For one thing, if $\sum_{i=1}^n x_i$ is not far from $\sum_{i=1}^n y_i$, we can take $r=0$ and the left-hand side is close to the right-hand side.
Example: $x_i=\frac{1}{(n-i)^2}$ and $y_i=\frac{1}{n-i}$. The right-hand side is about $\log n$. For the left-hand side, if we take $r=\frac{1}{n}$, then the first sum is already $\frac{1}{n}+\frac{1}{n-1}+\dots+\frac{1}{\sqrt{n}}$, which is about $\frac{\log n}{2}$, so we can take $c=\frac12$.
