MathDB

3

Part of 1983 Bundeswettbewerb Mathematik

Problems(2)

every positive real number is the sum of nine triplex numbers

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

11/22/2022
A real number is called triplex if it has a decimal representation in which none of 00 and 33 different digit occurs. Prove that every positive real number is the sum of nine triplex numbers.
number theoryDigits
k inside a pentagon

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

11/22/2022
There are kk points in the interior of a pentagon. Together with the vertices of the pentagon they form a (k+5)(k + 5)-element set MM. The area of the pentagon is defined by connecting lines between the points of MM into sub-areas in such a way that it is divided into sub-areas in such a way that no sub-areas have a point on their interior of MM and contains exactly three points of MM on the boundary of each part. None of the connecting lines has a point in common with any other connecting line or pentagon side, which does not belong to MM. With such a decomposition of the pentagon, there can be an even number of connecting lines (including the pentagon sides) go out? The answer has to be justified.
combinatoricscombinatorial geometrypentagon