MathDB

1

Part of 1984 Bundeswettbewerb Mathematik

Problems(2)

s(k) = 1 + z + z^2 + ...+ z^k divisible by n

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

11/21/2022
The natural numbers nn and zz are relatively prime and greater than 11. For k=0,1,2,...,n−1k = 0, 1, 2,..., n - 1 let s(k)=1+z+z2+...+zk.s(k) = 1 + z + z^2 + ...+ z^k. Prove that: a) At least one of the numbers s(k)s(k) is divisible by nn. b) If nn and z−1z - 1 are also coprime, then already one of the numbers s(k)s(k) with k=0,1,2,...,n−2k = 0,1, 2,..., n- 2 is divisible by nn.
number theorydivisibledivides
2 player game with 2n-digit num divisible by 9

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

11/21/2022
Let nn be a positive integer and M={1,2,3,4,5,6}M = \{1, 2, 3, 4, 5, 6\}. Two persons AA and BB play in the following Way: AA writes down a digit from MM, BB appends a digit from MM, and so it becomes alternately one digit from MM is appended until the 2n2n-digit decimal representation of a number has been created. If this number is divisible by 99, BB wins, otherwise AA wins. For which nn can AA and for which nn can BB force the win?
number theorycombinatoricsgamewinning strategy