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Let \(r_1\), \(r_2\), \(r_3\), ..., and \(r_{20}\) be the roots of the equation \(z^{20}-19z+2=0\).

Find

\[r_1^{20}+r_2^{20}+r_3^{20}+\dots+r_{20}^{20}\]

\[\large{\left(\dfrac {b-c}{a}+\dfrac {c-a}{b}+\dfrac {a-b}{c}\right)\left(\dfrac {a}{b-c}+\dfrac {b}{c-a}+\dfrac {c}{a-b}\right)}\]

If \(a+b+c=0\), then find ...

If one of the two series below is equal to 2, what is the value of the other one?

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Powder is sprinkled on a plate. When the plate is vibrated at a certain frequency, the powder moves into a pattern.

What is happening where the powder gathers?

Clip excerpted ...

\[\large\ \sum _{ n=0 }^{ 1947 }{ \frac { 1 }{ { 2 }^{ n }+\sqrt { { 2 }^{ 1947 } } } } = \frac { A }{ \sqrt { { 2 }^{ B } } }. \]

The expression above holds true for some odd integer \(A\) and positive ...

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