If $z$ is a complex number satisfying $z+\frac 1z=1,$ calculate $z^{10}+\frac 1{z^{10}}.$

$$\text{If } x \geq \frac{a}{b} \text{ and } x \geq \frac{c}{d}$$ for some $a, c \in \mathbb{R}_{\geq 0}$ and $b, d \in \mathbb{Z}_{\geq 1}$, is it true that $x \geq \frac{a + c}{b + d}$?

Find the number of ordered pairs of integers $(x,y)$ such that $x^2+y^2-xy=37$. I've never solved diophantine equations of degree 2 so I didn't know how to approach this problem. So I think there's no point in showing my efforts. All I did was to trial and error to find solutions but even that di...

I need a proof for this: Let $w$ be a primitive unity root order $3$. We need to prove that $(1-w +w^2)*(1+w-w^2) = 4$ Thanks.