Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^{2}+1\right)=P(x)^{2}+1$ for all real $x$.

Answer: For example, $P(x)=x, P(x)=x^{2}+1$ and $P(x)=x^{4}+2 x^{2}+2$.
Let $Q(x)=x^{2}+1$. Then the equation that $P$ must satisfy can be written $P(Q(x))=Q(P(x))$, and it is clear that this will be satisfied for $P(x)=x, P(x)=Q(x)$ and $P(x)=Q(Q(x))$.