Roots of Unity

Roots of Unity: Level 5 Challenges


Let a=2π2017a = \dfrac{2\pi}{2017} . Evaluate 1i<j2016(cosia)(cosja)\large \displaystyle \sum_{1 \leq i < j \leq 2016 } (\cos ia)(\cos ja) .

312α1+312α2+312α3+312α4\large \frac{31}{2- \alpha_1}+ \frac{31}{2- \alpha_2}+\frac{31}{2- \alpha_3} +\frac{31}{2- \alpha_4}

Given that 1,α1,α2,α3,α41,\alpha_1, \alpha_2, \alpha_3,\alpha_4 are distinct fifth roots of unity, evaluate the expression above.

k=120(α3k+β3k+ξ3k)\large \sum _{ k=1 }^{ 20 }{ \left( { \alpha }^{ 3k }+{ \beta }^{ 3k }+{ \xi }^{ 3k } \right) }

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

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

If x2x+1=0 x^{2}-x+1=0, then find the value of (x1x)2+a=12015(xa+1xa)2\left ( x-\frac{1}{x} \right )^{2}+\sum_{a=1}^{2015} \left ( x^a+\frac{1}{x^a} \right )^{2}

If w=e2πi/2015,w=e^{2\pi{i}/2015}, find k=1201411+wk+w2k.\sum_{k=1}^{2014}\frac{1}{1+w^k+w^{2k}}.

The equation x3=1x^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 x12=1 \large x^{\frac{1}{\sqrt{2}}}=-1

Note: When xx is a complex number different from 00, and rr is a real number, xrx^r can have more than one possible value. In this case, we assume that the complex number xx is a solution of the equation xr=s,x^r=s, where ss is a given real number, if at least one of the values of xrx^r is equal to s.s.


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