MathDB

1

Part of 1976 Bundeswettbewerb Mathematik

Problems(2)

9 lattice points in 3D, at least one midpoint is lattice point

Source: 1976 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

11/20/2022
Nine lattice points (i.e. with integer coordinates) P1,P2,...,P9P_1,P_2,...,P_9 are given in space. Show that the midpoint of at least one of the segments PiPjP_iP_j , where 1≤i<j≤91 \le i < j \le 9, is a lattice point as well.
combinatoricscombinatorial geometrylattice points
Bundeswettbewerb Mathematik 1976 Problem 2.1

Source: Bundeswettbewerb Mathematik 1976 Second round

9/11/2022
Prove that if nn is an odd natural number, then 1n+2n+⋯+nn1^n +2^n +\cdots +n^n is divisible by n2n^2.
number theoryDivisibility