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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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