Algebra

Algebraic Manipulation

Algebraic Manipulation: Level 2 Challenges

         

If ab=4a-b=4 and ab=45ab=45, what is the value of a3b3a^3 -b^3?

Given that aa, bb and cc are real numbers such that abc=1abc=1, find the value of (aab+a+1+bbc+b+1+cca+c+1)2.\left(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}\right)^2.

1+3+5+7++1992+4+6+8++200= ? \large \frac{1 + 3 + 5 + 7 +\ldots+ 199}{2 + 4 + 6 + 8 +\ldots+ 200} = \ ?

12+22+32++20122+20132+201421+2+3++2012+2013+2014= ?\large \frac{1^2+2^2+3^2+\cdots+2012^2+2013^2+2014^2}{1+2+3+\cdots+2012+2013+2014} = \ ?

Hint: 12+22+32++n2=16n(n+1)(2n+1) 1^2 + 2^2 + 3^2 + \cdots + n^2= \frac16 n(n+1)(2n+1) .

{x2y2=42xyx+y=2 \large { \begin{cases} {x^2-y^2=4-2xy} \\ { x+ y = 2 } \end{cases} }

If xx and yy satisfy the system of equations above, find the value of xyx-y.

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