Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice point in the interior?

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

Prove no three lattice points in the plane form an equilateral triangle.

Click for solution Here is a different approach leading to another generalization: Assume that triangle $ ABC$ is a lattice equilateral triangle. It is known that if $ \omega$ is the Brocard angle of a triangle $ ABC$, then $ \cot{\omega} = \frac {a^{2} + b^{2} + c^{2}}{4S}$. By Pick's theorem, the area of triangle $ ABC$ is (at least) rational. Thus, $ \cot{\omega} \in \mathbb{Q}$. But since $ ABC$ is equilateral, its Brocard angle is $ \frac {\pi}{6}$, and therefore $ \cot{\omega} = \cot{\frac {\pi}{6}} = \sqrt {3} \not\in \mathbb{Q}$. Contradiction. Generalization (communicated by Cezar Lupu): The Brocard angle of a lattice triangle is not a rational multiple of $ \pi$. This now follows from the fact that $ \cot{\mathbb{Q}\pi} \in \left\{ - 1, 0, 1\right\}$, combined with Johnson's theorem (which says that the Brocard angle of a triangle is less or equal $ \pi/6$, with equality when $ ABC$ is equilateral - see here: http://mathworld.wolfram.com/BrocardAngle.html).

The sidelengths of a polygon with $1994$ sides are $a_{i}=\sqrt{i^2 +4}$ $ \; (i=1,2,\cdots,1994)$. Prove that its vertices are not all on lattice points.

A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is $\frac{1}{2}$.

Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

Prove that on a coordinate plane it is impossible to draw a closed broken line such that coordinates of each vertex are rational, the length of its every edge is equal to $1$, the line has an odd number of vertices.

Prove that if a lattice parallellogram contains an odd number of lattice points, then its centroid.

Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity.

Prove that if a lattice parallelogram contains at most three lattice points in addition to its vertices, then those are on one of the diagonals.

Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.