Algebraic Identities

Calculate $99^2$.

Find real numbers $a,b,c$ such that $a^2+b^2+c^2+6=2(a+2b+c)$.

Find real numbers $x$ and $y$ such that $(x+y)^2=(x+1)(y-1)$.

Factorize $x^2-25$.

Calculate $101^3$.

Prove that $\sqrt[3]{\sqrt5+2}+\sqrt[3]{\sqrt5-2}$ is irrational.

Factorize $27a^9+8$.

Given $a+b=s$ and $ab=p$, find $a^3-b^3$ in terms of $s$ and $p$.

Assume $\frac1a+\frac1b+\frac1c=5$, $a+b+c=2$, and $abc=\frac14$. Find $a^3+b^3+c^3$.

Factorize $(x-y)^3+(y-z)^3+(z-x)^3$.

Assume $a,b,c$ are distinct real numbers. Prove that $\sqrt[3]{a-b}+\sqrt[3]{b-c}+\sqrt[3]{c-a}\neq0$.