Putnam

Algebra Difficulty 7.8 National Olympiad, round 2 Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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 $\lfloor 2a \rfloor \leq 2m+1$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.