Olympiad problems

PEN R Problems, 4

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

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.

greatest common divisor geometry number theory the geometry of numbers analytic geometry

PEN R Problems, 11

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.

geometry the geometry of numbers parallelogram

PEN R Problems, 5

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

geometry the geometry of numbers

PEN R Problems, 12

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

calculus integration the geometry of numbers analytic geometry

PEN R Problems, 6

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.

geometry the geometry of numbers

PEN R Problems, 1

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

induction the geometry of numbers mathematical induction

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]

geometric transformation geometry the geometry of numbers invariant analytic geometry coordinate method invariants modular arithmetic parity

PEN R Problems, 2

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

tetrahedron modular arithmetic vector geometry 3d geometry the geometry of numbers solid geometry vectors

PEN R Problems, 9

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