In order to get to Mars you must win a video game. The video game chooses $10$ points $(a_i,b_i)$ where $a_i$ and $b_i$ are single-digit integers, and places a disk with radius $1/3$ on each of the points. You must find a polynomial $f$ such that the graph of $f$ hits all $10$ discs.
However, you must choose your polynomial before seeing where the disks are. Find a polynomial that guarantees you a trip to mars.
The only clue I have for this problem is that for some point $x=h$ I must have a polynomial which is quite "steep" at that point and resembles the graph of a vertical line.In this way I am guaranteed that any other lattice point $(a_i,b_i)$ above it is included.
The only way I can imagine a polynomial doing that is when I have a product of positive expressions, i.e for $x=h$ I would have something like $(x-h)(x+r)(x+d)$ where $r,d$ are some large numbers.
The main conceptual difficulty I am facing here is to produce this kind of behavior for every single digit integer on the number line...