Algebra

Algebraic Manipulation: Level 3 Challenges

$S = \displaystyle\sqrt{9 - \sqrt{\dfrac{13}{9} + \sqrt{\dfrac{13}{81} - \sqrt{\dfrac{13}{6561} + \sqrt{\dfrac{13}{43046721} - .......}}}}}$

If $$S = \sqrt{\dfrac{a}{b}}$$ where $$a, b$$ are both primes, find $$a + b$$.

$\large 2016^{x}+2016^{-x}=3$

$\large \sqrt{\frac{2016^{6x}-2016^{-6x}}{2016^{x}-2016^{-x}}} = \, ?$

\Large\begin{align} \color{red}{x}\color{blue}{y} &= 7 \\ \color{red}{x}+\color{blue}{y} &= 5 \\ \color{red}{x}^3+\color{blue}{y}^3 &= \ \color{green}{?} \end{align}

$\sum_{j=2}^{2016} \sum_{k=1}^{j-1} \dfrac kj = \frac{1}{2}+\frac{1}{3}+\frac{2}{3}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\cdots+\frac{2013}{2016}+\frac{2014}{2016}+\frac{2015}{2016}= \, ?$

$\large {\begin{cases} x+y+z&=15 \\ xy+yz+xz&=72 \end{cases} }$

Let $$x,y,$$ and $$z$$ be real numbers satisfying the system of equations above.

Find the possible range of $$x$$.

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