AIME 1998 · 第 5 题

Solution

Though the problem may appear to be quite daunting, it is actually not that difficult. $\frac {k(k-1)}2$ always evaluates to an integer (triangular number), and the cosine of $n\pi$ where $n \in \mathbb{Z}$ is 1 if $n$ is even and -1 if $n$ is odd. $\frac {k(k-1)}2$ will be even if $4|k$ or $4|k-1$, and odd otherwise.

So our sum looks something like:

$\left|\sum_{i=19}^{98} A_i\right| =- \frac{19(18)}{2} + \frac{20(19)}{2} + \frac{21(20)}{2} - \frac{22(21)}{2} - \frac{23(22)}{2} + \frac{24(23)}{2} \cdots - \frac{98(97)}{2}$

If we group the terms in pairs, we see that we need a formula for $-\frac{(n)(n-1)}{2} + \frac{(n+1)(n)}{2} = \left(\frac n2\right)(n+1 - (n-1)) = n$. So the first two fractions add up to $19$, the next two to $-21$, and so forth.

If we pair the terms again now, each pair adds up to $-2$. There are $\frac{98-19+1}{2 \cdot 2} = 20$ such pairs, so our answer is $|-2 \cdot 20| = \boxed{040}$.