Algebra

Algebraic Manipulation

Algebraic Manipulation: Level 3 Challenges

         

Recall that
ex=1+x1!+x22!+x33!+. e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots. Then what is the value of 23!+45!+67!+21!+43!+65!+?\large \frac{\frac{2}{3!} + \frac{4}{5!} + \frac{6}{7!} + \cdots}{\frac{2}{1!} + \frac{4}{3!} + \frac{6}{5!} + \cdots}\, ?


Notation: !! is the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

{a+b+c=503a+bc=70\large \begin{cases} a+b+c=50 \\ 3a+b-c=70 \end{cases}

a,ba,b and cc are positive numbers satisfying the system of equations above.

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

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

20166x20166x2016x2016x=?\large \sqrt{\frac{2016^{6x}-2016^{-6x}}{2016^{x}-2016^{-x}}} = \, ?

S=9139+1381136561+1343046721.......S = \displaystyle\sqrt{9 - \sqrt{\dfrac{13}{9} + \sqrt{\dfrac{13}{81} - \sqrt{\dfrac{13}{6561} + \sqrt{\dfrac{13}{43046721} - .......}}}}}

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

j=22016k=1j1kj=12+13+23+14+24+34++20132016+20142016+20152016=?\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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