MathDB

2

Part of 1978 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1978 Problem 1.2

Source: Bundeswettbewerb Mathematik Round 1

10/12/2022
A set of n2n^2 counters are labeled with 1,2,…,n1,2,\ldots, n, each label appearing nn times. Can one arrange the counters on a line in such a way that for all x∈{1,2,…,n}x \in \{1,2,\ldots, n\}, between any two successive counters with the label xx there are exactly xx counters (with labels different from xx)?
combinatoricsarrangementLine
Bundeswettbewerb Mathematik 1978 Problem 2.2

Source: Bundeswettbewerb Mathematik 1978 Round 2

10/21/2022
Seven distinct points are given inside a square with side length 1.1. Together with the square's vertices, they form a set of 1111 points. Consider all triangles with vertices in M.M.
a) Show that at least one of these triangles has an area not exceeding 1\slash 16.
b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than 1\slash 16.
geometrysquareareaVerticesTriangle