2007 Alabama ARML TST Problems/Problem 12

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Problem

If $w^{2007}=1$ and $w\neq 1$, then evaluate

\[\dfrac{1}{1+w}+\dfrac{1}{1+w^2}+\dfrac{1}{1+w^3}+\cdots +\dfrac{1}{1+w^{2007}}.\]

Express your answer as a fraction in lowest terms.

Solution

For all $k \in \overline{1,2007}$:

$x = w^k$ is a root of $f(x) = x^{2007} - 1$.

$x = 1+w^k$ is a root of $g(x) = f(x-1) = (x-1)^{2007} - 1$.

$x = \dfrac{1}{1+w^k}$ is a root of $h(x) = g\left(\dfrac{1}{x}\right) = \left(\dfrac{1}{x} - 1\right)^{2007} - 1$.

$x = \dfrac{1}{1+w^k}$ is a root of $p(x) = x^{2007}h(x) = (1-x)^{2007} - x^{2007}$, since $\dfrac{1}{1+w^k} \neq 0$.


Since the degree of $p(x)$ is $2007$, $p(x)$ has exactly $2007$ roots. So, the problem is asking for the sum of the roots of $p(x)$.

Using the Binomial Theorem, $(1-x)^{2007} = -x^{2007} + 2007x^{2006} + \ldots + 1$.

So, $p(x) = (1-x)^{2007} - x^{2007} = -2x^{2007} + 2007x^{2006} + \ldots + 1$.

Therefore, by Vieta's Formulas, the sum of the roots of $p(x)$ is $-\dfrac{2007}{-2} = \boxed{\dfrac{2007}{2}}$.

See also

2007 Alabama ARML TST (Problems)
Preceded by:
Problem 11
Followed by:
Problem 13
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