Problem

Source: CSEMO 2005-7

Tags: arithmetic sequence, algebra unsolved, algebra



(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$, where $x \in R$ and $a$ is parameter. (2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$, and satisfy the condition \[ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. \] Find the maximum value of $n$.