Putnam 2005, Problem B-1

Find a nonzero polynomial $P(x,y)$ such that $P([a], [2a]) = 0$ for all real numbers $a$. (Note: $[v]$ is the greatest integer less than or equal to $v$.)

Solution:
Take \(P(x,y) = (y - 2x)(y - 2x - 1)\). To see that this works, first note that if \(m = \lfloor a \rfloor\), then \(2m\) is an integer less than or equal to \(2a\), so \(2m \leq \lfloor 2a \rfloor\). On the other hand, \(m + 1\) is an integer strictly greater than \(a\), so \(2m + 2\) is an integer strictly greater than \(2a\), so \(\lfloor 2a \rfloor \leq 2m + 1\).