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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.

# Algebraic Manipulation - Identities

If $$x$$ and $$y$$ are positive integers satisfying the equation $$3x+5y=90$$, what is the maximum possible value of $$x+y$$?

If the following equality always holds regardless of the value of $$k$$, what is $$x+y+z$$? $(k^2+k)x-(k^2-k)y-(k-3)z=30?$

If positive integers $$x$$ and $$y$$ satisfy the equation $$xy-5x-10y+37=0$$, the sum $$x+y$$ is constant. What is the value of $$x+y$$?

If $$a+b+c=1, a^2+b^2+c^2=11,$$ and $$abc=2,$$ what is the value of $a^2b^2+b^2c^2+c^2a^2?$

If $$x^8 = 829$$, what is $$(x-1)(x+1)(x^2+1)(x^4+1)$$?

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