This is a really nice question I found some days ago, so I translated it into English to share.
Suppose we have a water pipe which is infinitely long, with water flowing in it. We know that if a molecule of water in the pipe is at a point with the coordinate $x$, after $t$ seconds it will be at a point with the coordinate $P(t,x)$
Prove that if $P$ is a 2-variable polynomial, the speed of every water molecule is constant.
Here's what I did about the problem: We know that if a molecule is at $x$, after $t$ seconds it will be at $P(t,x)$ and after $s$ seconds it will be at $P(s,P(t,x))$ and also because it has moved for $s+t$ seconds, it is at $P(s+t,x)$ too. Thus, for every $s,t>0$ we have: $P(s,P(t,x))=P(s+t,x)$ so the two polynomials are equal because the equality must hold for infinitely many $s$ and $t$s. I don't know how to go any further so i would appreciate any help.
Feel free to edit the grammar.
Hint: Let $P(1,x) = R(x)$. This is a polynomial in one variable. Show by induction that $P(n, x) = R^{(n)}(x)$, i.e. $R$ iterated $n$ times. If $R(x)$ is a polynomial of degree $r$ and $Q(x)$ is a polynomial of degree $q$, what is the degree of $R(Q(x))$?