3

Part of 1991 Bundeswettbewerb Mathematik

Problems(2)

integer right triangle on lattice points has incenter a lattice point

Source: 1991 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

In a plane with a square grid, where the side length of the base square is

1

, lies a right triangle. All its vertices are lattice points and all side lengths are integer. Prove that the center of the incircle is also a lattice point.

geometryincenterlattice points

obtuse triangles

Source: Bundeswettbewerb Mathematik 1991, round two, problem 3

M

of points in the plane will be called obtuse, if any 3 points from

M

are the vertices of an obtuse triangle. a.) Prove: For each finite obtuse set

M

there is a point in the plane with the following property:

P

is no element from

M

M \cup \{P\}

is also obtuse. b.) Determine whether the statement from a.) will remain valid, if it is replaced by infinite.

geometry solvedgeometry