MathDB

Given a convex polygon with vertices at lattice points on a plane containing origin

O

. Let

V_1

be the set of vectors going from

O

to the vertices of the polygon, and

V_2

be the set of vectors going from

O

to the lattice points that lie inside or on the boundary of the polygon (thus,

V_1

is contained in

V_2

.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set

V_1

, and the second by the set

V_2

. Prove that there exists positive integer

c

that the following statement is true: if both grasshoppers can jump from

O

to some point

A

and the second grasshopper needs

n

jumps to do it, then the first grasshopper can use at most

n+c

jumps to do so.

Problem Statement