MathDB

4

Part of 2022 VJIMC

Problems(2)

nim-like game

Source: VJIMC 2022 1.4

4/11/2022
In a box there are 3131, 4141 and 5959 stones coloured, respectively, red, green and blue. Three players, having t-shirts of these three colours, play the following game. They sequentially make one of two moves: (I) either remove three stones of one colour from the box, (II) or replace two stones of different colours by two stones of the third colour. The game ends when all the stones in the box have the same colour and the winner is the player whose t-shirt has this colour. Assuming that the players play optimally, is it possible to decide whether the game ends and who will win, depending on who the starting player is?
combinatoricsgame
Additive property of a multiplicative function

Source: VJIMC 2022 Category II P4

4/9/2022
Let gg be the multiplicative function given by g(pα)=αpα−1,g(p^{\alpha}) = \alpha p^{\alpha-1}, for all α∈Z+\alpha\in\mathbb Z^+ and primes pp. Prove that there exist infinitely many integers nn such that g(n+1)=g(n)+g(1).g(n+1) = g(n) + g(1).
functionnumber theoryVJIMC