MathDB

3

Part of 1978 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1978 Problem 1.3

Source: Bundeswettbewerb Mathematik 1978 Round 1

10/12/2022
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.
number theoryDivisionSumpower of 2
game with 9 german words

Source: 1978 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

11/20/2022
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.
combinatorics