Problem

Source: 2021 Austrian Federal Competition For Advanced Students, Part 1 p4

Tags: game, combinatorics, game strategy



On a blackboard, there are 17 integers not divisible by 17. Alice and Bob play a game. Alice starts and they alternately play the following moves: Alice chooses a number a on the blackboard and replaces it with a2 Bob chooses a number b on the blackboard and replaces it with b3. Alice wins if the sum of the numbers on the blackboard is a multiple of 17 after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)