MathDB

2

Part of 2022 Middle European Mathematical Olympiad

Problems(2)

Shuffling cards

Source: MEMO 2022 I2

9/2/2022
Let nn be a positive integer. Anna and Beatrice play a game with a deck of nn cards labelled with the numbers 1,2,...,n1, 2,...,n. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if kk denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the kk topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.
combinatorics
Process on a blackboard again

Source: MEMO 2022 T2

9/2/2022
Let kk be a positive integer and a1,a2,...,aka_1, a_2,... , a_k be nonnegative real numbers. Initially, there is a sequence of n≥kn \geq k zeros written on a blackboard. At each step, Nicole chooses kk consecutive numbers written on the blackboard and increases the first number by a1a_1, the second one by a2a_2, and so on, until she increases the kk-th one by aka_k. After a positive number of steps, Nicole managed to make all the numbers on the blackboard equal. Prove that all the nonzero numbers among a1,a2,...,aka_1, a_2, . . . , a_k are equal.
algebra