Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.

Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated. [asy][asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy][/asy] The segments on the tiles define paths on the board. Determine the least possible number and the largest possible number of such paths. For example, there are $9$ paths on the $4\times4$ board shown below. [asy][asy]size(4cm);draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);draw((0,1)--(4,1));draw((0,2)--(4,2));draw((0,3)--(4,3));draw((1,0)--(1,4));draw((2,0)--(2,4));draw((3,0)--(3,4));draw((0,3.5)--(0.5,4),red);draw((0,2.5)--(1.5,4),red);draw((3.5,0)--(4,0.5),red);draw((2.5,0)--(4,1.5),red);draw((0.5,0)--(0,0.5),red);draw((2.5,4)--(3,3.5)--(3.5,4),red);draw((4,3.5)--(3.5,3)--(4,2.5),red);draw((0,1.5)--(1,2.5)--(1.5,2)--(0.5,1)--(1.5,0),red);draw((1.5,3)--(2,3.5)--(3.5,2)--(2,0.5)--(1.5,1)--(2.5,2)--cycle,red);[/asy][/asy]

Let $ABC$ be a triangle. The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$. Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.

A divisor $d$ of a positive integer $n$ is said to be a close divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?