MathDB

1

Part of 1992 Bundeswettbewerb Mathematik

Problems(2)

2 player game with stones, 1 bowl of p stones and one bowl of q stones

Source: 1992 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

11/20/2022
There are two bowls on the table, in one there are pp, in the other qq stones (p,q∈N∗p, q \in N* ). Two players AA and BB take turns playing, starting with AA. Who's turn: ∙\bullet takes a stone from one of the bowls ∙\bulletor removes one stone from each bowl ∙\bullet or puts a stone from one of the bowls into the other. Whoever takes the last stone wins. Under what conditions can AA and under what conditions can BB force the win? The answer must be justified.
combinatoricsgamegame strategy
m in positive integer n such that f(n) = 2n, f(1992) = 2199, f(2000) = 200

Source: 1992 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

11/20/2022
Below the standard representation of a positive integer nn is the representation understood by nn in the decimal system, where the first digit is different from 00. Everyone positive integer n is now assigned a number f(n)f(n) by using the standard representation of nn last digit is placed before the first. Examples: f(1992)=2199f(1992) = 2199, f(2000)=200f(2000) = 200. Determine the smallest positive integer nn for which f(n)=2nf(n) = 2n holds.
number theoryDigits