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# Roots of Unity

A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

# Roots of Unity: Level 5 Challenges

$\large \frac{31}{2- \alpha_1}+ \frac{31}{2- \alpha_2}+\frac{31}{2- \alpha_3} +\frac{31}{2- \alpha_4}$

Given that $$1,\alpha_1, \alpha_2, \alpha_3,\alpha_4$$ are distinct fifth roots of unity, evaluate the expression above.

$\large \sum _{ k=1 }^{ 20 }{ \left( { \alpha }^{ 3k }+{ \beta }^{ 3k }+{ \xi }^{ 3k } \right) }$

$\begin{cases} {f\left( -1 \right) =-9} \\ {f\left( 1 \right) =-7 } \\ {f\left( 3 \right) =19} \end{cases}$

If $$f\left( x \right)$$ is monic cubic polynomial having roots $$\alpha ,\beta ,\xi$$. Then evaluate topmost expression modulo 17.

If $$x^{2}-x+1=0$$ , then find the value of $\left ( x-\frac{1}{x} \right )^{2}+\displaystyle \sum_{a=1}^{2015} \left ( x^a+\frac{1}{x^a} \right )^{2}$

If $$w=e^{2\pi{i}/2015},$$ find $\sum_{k=1}^{2014}\frac{1}{1+w^k+w^{2k}}.$

The equation $$x^3=-1$$ has three solutions, one of which is real and the other two are non-real complex numbers. Determine the number and type of solutions of $\large x^{\frac{1}{\sqrt{2}}}=-1$

Note: When $$x$$ is a complex number different from $$0$$, and $$r$$ is a real number, $$x^r$$ can have more than one possible value. In this case, we assume that the complex number $$x$$ is a solution of the equation $$x^r=s,$$ where $$s$$ is a given real number, if at least one of the values of $$x^r$$ is equal to $$s.$$

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