Geometry of lattice points

All three vertices of an equilateral triangle cannot be lattice points.

How thick must the trunks of the trees in a regularly spaced circular forest grow if they are to block completely the view from the center?

Let $s$ be a given positive integer. Let every lattice point $p,q$ that satisfies the inequality $1\leq p^2+q^2\leq s^2$ be the center of a circle of radius $r$. If $r$ is sufficiently small, there exist rays running from the point 0, 0 to infinity without meeting the circles described above (the forest is transparent); such rays no longer exist if $r$ is sufficiently large (for $r=\tfrac 12$ the circles touch). Let $r=\rho$ be the value of $r$ which divides the two cases (limit of transparency). Then we have

In the plane draw the widest possible infinite straight ``path'', contained between two parallels and avoiding all the lattice points, that makes an angle equal to $\arctan x$ with the $y$-axis. Denoting the width of this path by $\varphi (x)$, determine $f(x)=\varphi (x)\sqrt{1+x^2}$.

Two lattice points $x,y$ and $x',y'$ are termed congruent mod $n$, if

Among any $kn^2+1$ distinct lattice points there are always $k+1$ that are congruent mod $n$.

In the plane consider a domain of area (Jordan content) $F$. It may contain no lattice points, but can always be moved parallel to itself so that $[F]+1$ lattice points lie in it.

A polygon is equiangular, has $n$ sides, and all its vertices are lattice points. Then $n$ is either 4 or 8.

A regular polygon has $n$ sides and all its vertices are lattice points. Then $n=4$.

A regular polygon has $n$ sides and all its vertices are lattice points. Then $n$ is equal to 3, 4 or 6.

(continued). Give examples showing that the three cases $n=3,4$ and 6 can actually occur.

All the vertices of a regular polyhedron are lattice points. Then the polyhedron is either a cube or a tetrahedron or an octahedron.

(continued). Give examples showing that the three cases mentioned above can actually occur.