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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.

Identities - Basic

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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