This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 10

PEN R Problems, 6
Tags: the geometry of numbers , geometry
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.

PEN R Problems, 1
Tags: the geometry of numbers , induction
Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice point in the interior?

PEN R Problems, 11
Tags: parallelogram , the geometry of numbers , geometry
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.

PEN R Problems, 8
Tags: invariant , the geometry of numbers , geometry , analytic geometry , geometric transformation
Prove that on a coordinate plane it is impossible to draw a closed broken line such that [list][*] coordinates of each vertex are rational, [*] the length of its every edge is equal to $1$, [*] the line has an odd number of vertices.[/list]

PEN R Problems, 2
Tags: modular arithmetic , 3d geometry , the geometry of numbers , tetrahedron , vector , geometry
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

PEN R Problems, 4
Tags: the geometry of numbers , modular arithmetic
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.

PEN R Problems, 10
Tags: analytic geometry , greatest common divisor , the geometry of numbers , geometry , number theory
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.

PEN R Problems, 5
Tags: geometry , the geometry of numbers
A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is $\frac{1}{2}$.

PEN R Problems, 12
Tags: integration , calculus , analytic geometry , the geometry of numbers
Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.

PEN R Problems, 9
Tags: geometry , parallelogram , the geometry of numbers
Prove that if a lattice parallellogram contains an odd number of lattice points, then its centroid.