1998 AIME Problems/Problem 5
Problem
Given that find
Solution
Though the problem may appear to be quite daunting, it is actually not that difficult. always evaluates to an integer (triangular number), and the cosine of where is 1 if is even and -1 if is odd. will be even if or , and odd otherwise.
So our sum looks something like:
If we group the terms in pairs, we see that we need a formula for . So the first two fractions add up to , the next two to , and so forth.
If we pair the terms again now, each pair adds up to . There are such pairs, so our answer is .
See also
1998 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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