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## Algebraic Manipulation

Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.

# Level 3

$\large S = 2014^{3}-2013^{3}+2012^{3}-2011^{3}+\cdots+2^{3}-1^{3}$

What is the largest perfect square that divides $$S$$ above?

$\large x^{63} + x^{44} + x^{37} + x^{31} + x^{26} + x^9 + 6$

If $$x$$ satisfies the equation $$\left(x + \dfrac1x\right)^2 = 3$$, then find the value of the expression above.

$\large \begin{cases} a+b+c=50 \\ 3a+b-c=70 \end{cases}$

$$a,b$$ and $$c$$ are positive numbers satisfying the system of equations above.

If the range of $$5a + 4b + 2c$$ is $$(m,n)$$, what is $$m+ n$$?

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

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

Find the possible range of $$x$$.

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

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