MathDB

2

Part of 1992 Bundeswettbewerb Mathematik

Problems(2)

n = b + c + d +... =b * c * d * ... , good positive integer

Source: 1992 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

11/20/2022
A positive integer nn is called good if they sum up in one and only one way at least of two positive integers whose product also has the value nn. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.
number theorySumProduct
algorithm to create a seuqence of 2^n words created by n gifits of {0,1}

Source: 1992 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

11/20/2022
All nn-digit words from the alphabet {0,1}\{0, 1\} considered. These 2n2^n words should be in a sequence w0,w1,w2,...,w2−1w_0, w_1, w_2, ..., w_{2^-1} be arranged that wmw_m from wm−1w_{m-1} by changing of a single ornament (m=1,2,3,...,2n−1m = 1, 2, 3, ..., 2n-1). Prove that the following algorithm achievesthis : a) Start with w0=000...00w_0 = 000... 00. b) Let wm−1=a1a2a3...anw_{m-1} = a_1a_2a_3 ... a_n with ai∈{0;1}a_i \in \{0; 1\}, i=1,2,3,...,ni = 1, 2, 3, ..., n. Determine the exponent e(m)e(m) of the highest power of two dividing mm and set j=e(m)+1j = e(m)+1. In wm−1w_{m-1} replace the ornament aja_j with 1−aj1-aj. this is now wmw_m.
algorithmcombinatoricsWords