Question 19
Let $a_{1}, a_{2}, \ldots, a_{2013}$ be real numbers satisfying the following conditions:
- $a_{1}=a_{2013}=0$
- $|a_{i}-a_{i+1}|\lt 1$, for $1 \leq i \leq 2012$.
- $\sum_{i=1}^{2013} a_{i}=0$.
Find the greatest possible value of the sum $\sum_{i=1}^{m} a_{i}$, where $m$ ($1 \leq m \leq 2013$) is allowed to vary, in addition to the sequence $\{a_{i}\}$.
- $a_{1}=a_{2013}=0$
- $|a_{i}-a_{i+1}|\lt 1$, for $1 \leq i \leq 2012$.
- $\sum_{i=1}^{2013} a_{i}=0$.
Find the greatest possible value of the sum $\sum_{i=1}^{m} a_{i}$, where $m$ ($1 \leq m \leq 2013$) is allowed to vary, in addition to the sequence $\{a_{i}\}$.