2
Part of 2023 All-Russian Olympiad
Problems(2)
Operation with binary strings
Source: All-Russian MO 2023 Final stage 9.2
4/23/2023
Initially, a word of letters with letters and letters is written on a blackboard. In each operation, we may choose a contiguous string of any length with equal number of letters and equal number of letters , reverse those letters and then swap each with and each with (Example: after the operation becomes ). Decide if it possible to choose initial word, so that after some operations, it will become the same as the first word, but in reverse order.
combinatorics
Averages can’t be pairwise distinct
Source: All-Russian MO 2023 10.2, 11.2
4/24/2023
A group of kids has a deck of cards numbered by . The first kid takes the deck, shuffles it, and then takes the cards one by one; when he takes a card (not the last one in the deck), he computes the average of the numbers on the cards he took up to that moment, and writes down this average on the blackboard. Thus, he writes down numbers, the first of which is the number on the first taken card. Then he passes the deck to the second kid which shuffles the deck and then performs the same procedure, and so on. This way, each of kids writes down numbers. Prove that there are two equal numbers among the numbers on the blackboard.
combinatoricsAverageilostthegame