Algebra

# Roots of Unity: Level 5 Challenges

Let $a = \dfrac{2\pi}{2017}$. Evaluate $\large \displaystyle \sum_{1 \leq i < j \leq 2016 } (\cos ia)(\cos ja)$.

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

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 $\left ( x-\frac{1}{x} \right )^{2}+\sum_{a=1}^{2015} \left ( x^a+\frac{1}{x^a} \right )^{2}$

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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