1. Is there any polynomial \(f\in\mathbb{Q}[x,y]\) such that \(f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}\) is a bijection?
  2. Is there any polynomial \(f\in\mathbb{Q}[x,y]\) such that \(f(\mathbb{Z} \times \mathbb{Z})=\mathbb{N}\)?
  3. Let \(f\) and \(g\) be two continuous mappings from \([0,1]^n\) to itself, such that \(f\circ g=g\circ f\). Does there always exist \(x\in [0,1]^n\) such that \(f(x)=g(x)\)?
  4. If \(2^x\) and \(3^x\) are integers, must \(x\) be as well?
  5. Let \(P,Q\in\mathbb{R}_{\geq 0}[x]\) be two monic polynomials such that \(PQ\in\{0,1\}[x]\). Is it always the case that \(P\in\{0,1\}[x]\)?