Dsquared

Die Ausgangssituation:

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Ich wohne dauerhaft in den Niederlanden, habe mich dort angemeldet, habe eine Sofi-Nummer, usw.

Ich arbeite zwar für eine deutsche Firma, aber ich mache diese Arbeit ausschliesslich von Zuhause in den Niederlanden (per Computer).

Ich habe eine Dauerarbeitsvertrag, also unbefristet.

Ich bin Engländer (also EG-Bürger).

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Ich weiss schon, dass ich in den Niederlanden vom Versicherungsbetrieb CZ versichert werden muss.

Es ist mir aber unklar, ob ich in Deutschland versichert bleiben muss. In diesem Zusammenhang aber bekam ich kein "E106"-Formular von meinem Versicherungsbetrieb (AOK) in Deutschland, als ich ihn informierte, ich würde in die Niederlande umziehen.

Weiss jemand, ob ich meine AOK-Versicherungsvertrag beenden muss?

Suppose there are n distinct nodes on the circumference of a circle, and that each node is connected to every other node by a straight line.

The only restriction on these nodes (apart from distinctness) is that they must be positioned such that no point inside the circle has more than two lines passing through it. (This is always possible, since for any random set of n distinct nodes, we can always "tweak" their positions a bit until the criterion is satisfied.)

How many regions of the circle are there (as a function of n)?

(Please provide proofs for your answers.)

This is a follow-up to:

http://answers.yahoo.com/question/index;_ylt=Apz2....

The result given in the original question stated the smallest value of N, such that (N+1) boxes cannot be labelled. This is Nm1 = 199991. However, looking at slightly higher numbers, it's clear that these numbers would be sufficient to label the boxes, e.g. N = 200000.

The question here, as a counterpart to the original, is what is the smallest value of N (call it Nm2), such that for ALL N > Nm2, it is not possible to label all N boxes?

(The same conditions as in the original question apply, e.g. 10 labels in each box, etc.)

Tipp: Keine rein mathematische Aufgabe, man muss ein bißchen quer denken.

Here r is any integer.

As a direct corollary of this result, show that

n*(1+r+r^2) = 180 has no solutions with n and r both integers and r > 1.

(This result provides an alternative, "non-number-crunching" approach to a recently-posed question on angles of a triangle forming a geometric progression (except that the possibility of r being rational is not considered in the present problem):

http://answers.yahoo.com/question/index;_ylt=AuR76... )