Consider the following problem: Given the following sets $$\begin{align}A_u&=\{x^2:x\in [u,2u-1],\exists s,t : x^2=s^3+2s^2+st+t\}\\B_u&=\{x^2:x\in [u,2u-1],\exists s,t : x^2=2s^3+2s^2+2st+t\}\end{align}$$ prove that $\exists N\in\Bbb Z^+$ such that $\forall u\gt N, |A_u|\ge|B_u|$. My ...