4

Part of 1983 Bundeswettbewerb Mathematik

Problems(2)

distance between 2 of n + 1 points is an integer

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

g

be a straight line and

n

a given positive integer. Prove that there are always n different points on g to choose as well as a point not lying on g in such a way that the distance between each two of these

n + 1

points is an integer.

distancecombinatoricscombinatorial geometry

Weird Recursion - Weird Explicit Formula

Source: BWM 1983, Round 2 - #4

f(0), f(1), f(2), \dots

be a sequence satisfying f(0) = 0 \text{and} f(n) = n - f(f(n-1)) for

n=1,2,3,\dots

. Give a formula for

f(n)

such that its value can be immediately computed using

n

without having to compute the previous terms.

recursionalgebraalgebra unsolvedSequence