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1978 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

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Bundeswettbewerb Mathematik 1978 Problem 1.2

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)?

Bundeswettbewerb Mathematik 1978 Problem 2.2

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.
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Bundeswettbewerb Mathematik 1978 Problem 1.3

For every positive integer nn, define the remainder sum r(n)r(n) as the sum of the remainders upon division of nn by each of the numbers 11 through nn. Prove that r(2k−1)=r(2k)r(2^{k}-1) =r(2^{k}) for every k≥1.k\geq 1.

game with 9 german words

Sunn and Tacks play a game alternately choosing a word among the following (German) words: ”bad”, ”binse”, ”kafig”, ”kosewort”, ”maitag”, ”name”, ”pol”, ”parade”, ”wolf”. Two words are said to compatible if they have exactly one consonant in common. In the first round, Sunn selects a word for herself and one for Tacks. In every consequent round, each player selects a word that is compatible with the one they chose in the previous round. Tacks wins the game if the two players successively select the same word. (a) Prove that Tacks can always win. How many rounds are necessary for that? (b) Upon Sunn’s desire, the word ”kafig” was replaced with the word ”feige”. Prove that Sunn can prevent Tacks from winning.
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