Difference between revisions of "2004 AIME I Problems/Problem 13"
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We see that the expression for the [[polynomial]] <math>P</math> is very difficult to work with directly, but there is one obvious transformation to make: sum the [[geometric series]]: | We see that the expression for the [[polynomial]] <math>P</math> is very difficult to work with directly, but there is one obvious transformation to make: sum the [[geometric series]]: | ||
− | <math>P(x) = (\frac{x^{18} - 1}{x - 1})^2 - x^{17} = \frac{x^{36} - 2x^{18} + 1}{x^2 - 2x + 1} - x^{17} = \frac{x^{36} - x^{19} - x^{17} + 1}{(x - 1)^2} = \frac{(x^{19} - 1)(x^{17} - 1)}{(x - 1)^2}</math>. | + | <math>P(x) = \left(\frac{x^{18} - 1}{x - 1}\right)^2 - x^{17} = \frac{x^{36} - 2x^{18} + 1}{x^2 - 2x + 1} - x^{17} = \frac{x^{36} - x^{19} - x^{17} + 1}{(x - 1)^2} = \frac{(x^{19} - 1)(x^{17} - 1)}{(x - 1)^2}</math>. |
This [[expression]] has [[root]]s at every 17th root and 19th root of unity, other than 1. Since 17 and 19 are [[relatively prime]], this means there are no duplicate roots. Thus, <math>a_1, a_2, a_3, a_4</math> and <math>a_5</math> are the five smallest fractions of the form <math>\frac m{19}</math> or <math>\frac n {17}</math> for <math>m, n > 0</math>. | This [[expression]] has [[root]]s at every 17th root and 19th root of unity, other than 1. Since 17 and 19 are [[relatively prime]], this means there are no duplicate roots. Thus, <math>a_1, a_2, a_3, a_4</math> and <math>a_5</math> are the five smallest fractions of the form <math>\frac m{19}</math> or <math>\frac n {17}</math> for <math>m, n > 0</math>. |
Revision as of 17:52, 2 December 2006
Problem
The polynomial has 34 complex roots of the form
with
and
Given that
where
and
are relatively prime positive integers, find
Solution
We see that the expression for the polynomial is very difficult to work with directly, but there is one obvious transformation to make: sum the geometric series:
.
This expression has roots at every 17th root and 19th root of unity, other than 1. Since 17 and 19 are relatively prime, this means there are no duplicate roots. Thus, and
are the five smallest fractions of the form
or
for
.
and
can both be seen to be larger than any of
, so these latter five are the numbers we want to add.
and so the answer is
.