1

Part of 1997 Bundeswettbewerb Mathematik

Problems(2)

15 integers from 100 with difference divisible by 7

Source: 1997 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

100

integers, is it always possible to choose

15

of them such that the difference of any two of the chosen numbers is divisible by

7

? What is the answer if

15

16

number theorydivisible

3 faced of regular tetrahedron are painted in white and one in black

Source: 1997 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

Three faces of a regular tetrahedron are painted in white and the remaining one in black. Initially, the tetrahedron is positioned on a plane with the black face down. It is then tilted several times over its edges. After a while it returns to its original position. Can it now have a white face down?

Coloringtetrahedroncombinatorics